5.5 The real zeros of a polynomial function

Size: px
Start display at page:

Download "5.5 The real zeros of a polynomial function"

Transcription

1 5.5 The real zeros of a polynomial function Recall, that c is a zero of a polynomial f(), if f(c) = 0. Eample: a) Find real zeros of f()= + -. We need to find for which f() = 0, that is we have to solve the equation + b b 4ac - = 0. Using quadratic formula we get a 4()( ) 5 The equation has two solutions, therefore f()= + has two zeros 5, 5 b) Find real zeros of f() = 3. To find zeros we need to solve the equation 3 = 0. Using the square root method we get = 3 and 3. Hence the zeros are 3 and 3. c) Find real zeros of f() = + 4. The equation + 4 = 0 has no real solution ( = -4 is never true, no matter what is). Therefore f() = + 4 has no real zeros. Theorem Division Algorithm for Polynomials Suppose that f() and g() are polynomials and that the degree of g is not zero (g() is not a constant). Then there are unique polynomials Q() and R() such that f ( ) g( ) R( ) Q( ) g( ) or f ( ) Q( ) g( ) R( ) where the degree of R() < the degree of g(). f() is called the dividend, g() is called the divisor, Q() is the quotient and R() is the remainder. If g() = (-c), then f() = (-c)q() + R(), and the degree of R() < the degree of g()=. Hence, R() is a constant, R() = R. Note also that when = c, we get f(c) = (c-c)q(c) + R = R. Theorem Remainder Theorem If f is a polynomial, then when f() is divided by (-c), the remainder R = f(c). Factor Theorem (-c) is a factor of a polynomial f() if and only if f(c) = 0 Proof: f() = (-c)q() + f(c) Eample: a) Let f() = Find the remainder when f() is divided by - When f () is divided by (-), then the remainder R = f() = 4() 4-5() - 4= = 0. Hence, (-) is a factor of f(), that is f() = (-)Q(), where Q() is a polynomial. b) Let f() = Find the remainder when f() is divided by +.

2 Note first that + = -(-). When f () is divided by ( (- )), then the remainder is R = f(-) = (-) 6 8(-) 4 + (-) 9= = -4 Therefore (+) is not a factor of f(). Theorem A polynomial of degree n can have at most n real zeros. Eample: How many real zeros can the polynomial f() = have? Degree of f() = 6, so there are at most 6 real zeros. Eample: 5 a) Since f() = + has two real zeros and factors of f(), and f ( ) 5, then 5 5 b) Similarly, f ( ) 3 3 ( 3) ( 3)( 3) 5 and 5 are c) f() = + 4 has no real zeros, so it cannot be factored over the real numbers. We call such a polynomial irreducible or prime Theorem Every polynomial function with real coefficients can be uniquely factored into a product of linear factors (-c) and /or irreducible quadratic factors (a + b+c) Theorem A polynomial function with real coefficients and with odd degree must have at least one real zero. Theorem Rational Zeros Theorem Supose f() = a n n + a n- n- + +a + a 0, a n, a 0 0, where all coefficients a k are integers, is a polynomial of degree greater than 0. If p/q is a rational zero of f(), then p divides a 0 and q divides a n. In other words, any rational zero of a polynomial with integer coefficients, must be of the form factor of a0 factor of a n

3 Eample: List all potential zeros of the polynomial f() = a 3 = -4 ; factors: ±, ±, ±4 (p) a 0 = 6 ; factors : ±, ±, ±3, ±6 (q) p q factor factor of of a 0 a n,, 3, 6,, 3,, Remark: A polynomial function with integer coefficients might not have any rational (or even real) zeros. For eample f() = - has no rational zeros (the zeros are ), and f() = + has no real zeros at all. Once we know potential zeros, we can check whether they are actual zeros, either by computing value of the polynomial at the potential zero or performing division. Eample: Find all real zeros of the given polynomial and then factor the polynomial f() = degree of f = 3; so there are at most 3 zeros all coefficients are integers, so we will look for rational zeros among the numbers of the form a 0 = - 0 ; factors of a 0 : ±, ±, ±4, ±5, ±0, ±0 a n = ; factors of a n : ± p factor of a0,, 4, 5, 0, 0 q factor of a n factor of a0 factor of a n (i) Is = a zero? f()= () 3 + 8() + () -0= = 0; So, = is a zero Therefore, f() = = (-)Q(). To find Q(), we must divide f() = by (-). We can use either long division or a simplified version called synthetic division (applied ONLY when the divisor is of the form (-c)) Long division: this is the quotient Q() divide 3 by 3 multiply (-) and subtract; divide 9 by 9 9 multiply 9(-) and subtract; 0-0 divide 0 by 0-0 multiply 0(-) and subtract 0 this is the remainder R

4 Synthetic division: So, Q() = and f() = = (-)( + 9+0). Now we must find zeros of Q() = Since it is a quadratic polynomial we solve the equation Q() = 0 or = 0 either by factoring or using the quadratic formula (if factoring does not work). Since = (+4)(+5), the equation becomes (+4)(+5) = 0 the zeros are -4 and -5. Therefore the zeros of f() are, -4, -5 and f() = (-)(+4)(+5) Solving polynomial equations If f() is a polynomial, then the equation f() = 0 is called a polynomial equation. To solve such an equation, means to find zeros of the polynomial function f(). Eample: Solve the equation = 0 Let f() = We look for the rational zeros, since the equation has integer coefficients. Factors of a o are : ±, ±, ±4, ±8 Factors of a n =a 3 are : ±, ± p factor of a0 Possible zeros: q factor of a n,, 4, 8, Is = a zero? f() = = 9 0, so = is not a zero Is = - a zero? f(-) = = - 5 0, so = - is not a zero Is = a zero? f() = = 0, so = is a zero Since = is a zero, then (-) is a factor of f(), that is f() = (-)Q(). To find Q() we ll use synthetic division And therefore, Q() = Since Q() is a quadratic polynomial, we use the learned techniques to try to factor it. Indeed Q() = -7-4 = (+)(-4). Therefore f() = = (-)( -7-4)= (-)(+)( - 4). So the zeros, or the solutions of f() = 0, are, -/, 4.

5 5. Polynomial Functions A polynomial functions is a function of the form f() = a n n + a n- n- + + a + a 0 Eample: f() = The domain of a polynomial function is the set of all real numbers. The -intercepts are the solutions of the equation a n n + a n- n- + + a + a 0 = 0 The y-intercept is y = f(0) = a 0. The graph of a polynomial function does not have holes or gaps (we say that a polynomial function is continuous) and it does not have sharp corners or cusps (we say it is smooth). The graphs of some power functions are given below Power functions: f() = n, n is a natural number n- even n- odd Notice that when n is even, a power function y = n behaves like a parabola (graph is symmetric about the y-ais and contains points (-,), (0,0), (,)). When n is odd, a power function y = n, n > has the graph similar to the cube function (symmetric about the origin, contains the points (-,-), (0,0), (,)). Power function f() = a n, a 0 The graph of f() = a n is obtained from the graph of y = n by stretching by a factor of a, if a is positive, and stretching by the factor of a and reflecting about the -ais, if a is negative. n- even n odd

6 Zeros of a polynomial function If r is such a number that f(r) = 0, then r is called a zero of the function f. If r is a zero of a polynomial function f, then we have the following: (i) f(r) = 0 (ii) (r, 0 ) is an -intercept of f (iii) (-r) is a factor of f(), that is f() = (-r)q() ( q() is the quotient in the division f() (-r)) We say that r is a zero of multiplicity n, if n is the largest power, such that f() = (-r) n q() Eample: Let f() = Note that f() = () 4 -() 3 +3() -()- = = 0. Therefore, r = is a zero of f. This also means, that (,0 ) is an -intercept and that (-) is a factor of f(), that is, when f() is divided by (-) the remainder is 0. To find the other factor, q, we perform the division Therefore, f() = = (-)( ). What is the multiplicity of that zero? Since (-) is a factor of f(), then the multiplicity of = is at least. If = had the multiplicity two, then f() would have (-) as a factor. Which means that the quotient q above, q() = 3 + +, would have (-) as a factor. If (-) were a factor of q(), then = would be a zero of q. But q() = () 3 () + () + = 3 0. This means that (-) is not a factor of q() and consequently, (-) is not a factor of f(). Hence, the multiplicity of = is one. Eample: Find all zeros of function f() = (-3)(+)(+3) (-5) 4 and determine their multiplicity To find zeros, solve the equation f() = 0 (-3)(+)(+3) (-5) 4 = 0 (use the Zero Product property) -3 = 0 or + = 0 or +3= 0 or - 5 = 0 So the zeros are : 3, -, -3, 5 To find the multiplicities: (i) factor the polynomial completely; use eponents to indicate multiple factors (ii) The multiplicity of a zero r is the eponent of the factor (-r) that appears in the product

7 f() = (-3) (+) (+3) (-5) 4 zeros multiplicity 4 Suppose r is the zero of a polynomial function. (Remember that (r,0) is then an -intercept) If r is a zero of even multiplicity, then the graph of f will touch the ais at the intercept (r, 0) as shown below. or If r is a zero of odd multiplicity, then the graph of f will cross the -ais at the intercept (r,0) as shown below. Multiplicity one Multiplicity larger than (odd) or or

8 One of the theorems of algebra says that every polynomial can be factored in such a way that the only factors are : (a) a number (the leading coefficient) (b) ( r) n, where r is a zero with multiplicity n (c) ( +b +c) m, where + b + c is prime Remark: This theorem says that such factorization is possible, but it does not say how to obtain such factorization. End behavior of a polynomial function When is large ( is large positive or large negative), then the graph of f() = a n n + a n- n- + + a + a 0 behaves like the graph of y = a n n, where a n is the leading coefficient and n is the degree of f(). n- odd a n positive a n negative n- even a n positive a n negative Eample: Determine the degree and the leading coefficient of the polynomial function f() = 3(-)(+4) ( +) 3. Give the equation of the power function that the function f behaves like for with large absolute value. f() = 3(-)(+4) ( +) 3 This polynomial is already factored. The leading coefficient is 3. The degree can be obtained by adding the highest eponents of from each factor

9 factor 3 - (+ ) ( +) 3 degree 0 6 Degree = = 9 Therefore, for large, f() behaves like y = 3 9. Sketching the graph of a polynomial function (I) use transformations, when possible (II) If the transformations cannot be performed, use the information above to sketch the graph Eample: Graph f() = (-) We can graph this function using transformations The order of transformations is as follows ) graph basic function y = 5 ) Shift the graph unit to the right to obtain y = (-) 5 3) Stretch the graph by a factor of, to obtain y = (-) 5 4) Shift the graph up by 3 units to obtain y = (-) 5 + 3

10 Eample: Graph f() = (-3)(+4) 3 Transformations cannot be used (i) Determine the zeros, if any, of f, their multiplicity and the behavior of graph near each zero f() = (-) (+) 3 zeros - multiplicity 3 behavior of graph touches -ais at Crosses -ais at - like a cubic function (ii) Determine the end behavior of f We need the leading coefficient and the degree of f() f () = (-3)(+4) 3 Leading coefficient = Degree: factor (-3) (+4) 3 degree 0 3 Degree of f = 0++3 = 5 f() behaves as y = 5 for large (iii) Use the information from (i) and (ii) to draw the graph. The graph will start in the third quadrant (green piece). It will continue to the first zero (-) at which it will cross the - ais like a cubic function (red piece). The graph will increase for a while, but then it will have to turn to reach the second zero( ), at which it will touch the -ais (red piece). Since there are no more zeros, the graph will continue, eventually to reach green piece that depicts its behavior for the large positive.

11 Eample: Graph f() = ( +)(+4) Note that + is always positive, so f() can be zero only when = 0 or = -4 zeros 0-4 multiplicity behavior of graph touches -ais at Touches ais at - 4 End behavior: Leading coefficient = factor + (+4) degree Degree of f() = + + = 6 and f behaves like y = 6 for large, that is, it looks as below

12 The graph of the function is below Remarks: - We don t have enough information to know at what points eactly the graph will change the direction (these points are called turning points). (You can learn this in Calculus.) But, we know that there are at most (n-) turning points (n is the degree of the polynomial) - Though we know how the graph of a polynomial function behaves for large positive and negative values and close to its zeros, we need Calculus to determine its behavior in- between. In this course however, we ll assume that nothing etraordinary takes place there. 5. Properties of Rational functions A rational function is a function of the form 0 0 ) ( ) ( ) ( b b b b a a a a q p polynomial polynomial f k k k k n n n n Eample ) ( 4 f The domain of a rational function is the set of all real numbers ecept those, for which q() = 0 To find the domain: (i) solve q() = 0 (ii) Write Df = { q() 0} Eample: Find the domain of ) ( 4 f (i) Solve: denominator = 0

13 ( )() 4 0 ( ) 4 5 ( 5) (ii) Df = { 5 } = (, 5) ( 5, 5) ( 5, ) 5 Recall the graphs of f ( ) and g( ) f ( ) Note that when is small (close to 0 on the -ais) then its reciprocal is a large number (positive when is positive and negative when is negative). For eample if = then / = 0, 000; if = then / = -,000,000. The smaller, the larger /. In mathematics we indicate that fact by saying that when > 0 approaches 0, then / = f() increases without bound and write as 0 +, then f() + or lim f ( ). Note that in such a case, the point (, f()) on the graph approaches 0 the y-ais. We say that the y-ais is a vertical asymptote. If on the other hand + (that is, increases without bound) than the values / become smaller and smaller (if = 0,000, then / = 0.000; if =,000,000 then / = ) so we say that as +, then / = f() 0 or that lim f ( ) 0. Note that in this case, the point (, f()) on the graph of the function approaches the -ais. We say that the -ais is a horizontal asymptote.

14 g( ) A rational function often has asymptotes: vertical and/or horizontal/oblique. Informally speaking, an asymptote is a straight line (vertical, horizontal or slanted) toward which the graph comes near. Vertical and horizontal asymptotes vertical and oblique asymptotes vertical and horizontal asymptotes Eample: Given the graph of a function, find the domain and give the equations of any asymptotes a) Domain ={ -,3} ; vertical asymptotes: =-, = 3; horizontal asymptote: y =

15 b) Domain = { } ; vertical asymptote: = ; slanted asymptote : y = + ( line passes through (0,) and (,5)) How to find asymptotes Vertical:. Reduce f() to the lowest terms: (i) factor completely the numerator and the denominator; (ii) cancel common factors. Solve the equation: denominator = 0 3. If = r is a solution found in, then the line = r is a vertical asymptote Horizontal: a) if the degree of the numerator < the degree of the denominator, then the line y = 0 is the horizontal asymptote an (b) if the degree of the numerator = the degree of the denominator, then the line y = is b the horizontal asymptote (c) if the degree of the numerator > the degree of the denominator, then the graph does not have a horizontal asymptote, however k Oblique: (d) if the degree of the numerator = + the degree of the denominator, then the line y = (quotient obtained by dividing the numerator by the denominator) is an oblique(slanted) asymptote. Remarks:. A rational function can have only one horizontal/oblique asymptote, but many vertical asymptotes.. If a rational function has a horizontal asymptote, then it does not have an oblique one. 3. The graph of a rational function can cross a horizontal/oblique asymptote, but does not cross a vertical asymptote 4. Horizontal/oblique asymptotes describe the behavior of function for with large absolute value (the end behavior); vertical asymptotes describe the behavior of function near a point.

16 Eample: Find the asymptotes for the following functions a) 3 5 f ( ) 6 Vertical asymptote: ) f is in lowest terms Horizontal/oblique asymptote: ) -6 = 0 = 6 = 3 3) vertical asymptote: = 3 degree of numerator () = degree of the denominator(), 3 y is the horizontal asymptote 5 b) f ( ) Vertical asymptote: ) ( 5 5 f ) is in lowest terms (numerator can t be factored) ( ) ) = 0 3 (-) = 0 = 0 or - = 0 = 0 or = 3) vertical asymptotes : = 0, = Horizontal/oblique asymptote: degree of numerator () < degree of the denominator(3), y = 0 is the horizontal asymptote c) f 5 3 ) ( Vertical asymptote: ) f() is in lowest terms (the denominator cannot be factored) ) + = 0 = - (not possible) no solution 3) vertical asymptotes : none Horizontal/oblique asymptote: degree of numerator (5) > degree of the denominator(), there is no horizontal asymptote degree of numerator (5) + degree of the denominator(), there is no oblique asymptote d) f 3 ) 3 ( 4 Vertical asymptote: ) f() is in lowest terms ) - = 0 = =

17 3) vertical asymptotes : = -, = Horizontal/oblique asymptote: degree of numerator (3) > degree of the denominator(), there is no horizontal asymptote degree of numerator (3) = + degree of the denominator(), there is an oblique asymptote Oblique asymptote: y = 3-4 Remark: When looking for a vertical asymptote, it is important to make sure that the function is reduced to the 4 lowest terms. To see why, consider function f ( ). Note that 4 ( )( ) f ( ) ( ), ( ) hole at (-, -4), hence has no asymptote. if. The graph of f() is therefore a straight line y = -, with a Graphing rational functions We can graph some rational functions using transformations Eample: Use transformations to graph f ( ) ( 3) Basic function: y y (shift left by 3) ( 3)

18 y ( 3) ( 3) (vertical stretch) y ( 3) (Shift down by ). Find the domain: (i) solve q() = 0 (ii) Df = { q() 0} 5.3 Sketching the graph of a rational function f ( ) p( ) q( ). Find - and y-intercepts: y-intercept: y = f(0) - intercepts: numerator = 0 3. Find vertical asymptotes, if any Remark: If = r is ecluded from the domain and = r is not a vertical asymptote, then the graph of f will pass through the point (r, reduced f (r)) but the point itself will not be included. We put an open circle around that point The graph of f has a hole at = r 4. Find the horizontal/oblique asymptote, if any. 5. Find the points where the graph crosses the horizontal/oblique asymptote y = m +b (i) solve the equation f() = m + b 6. Check for symmetries (i) If f(-) = f(), then the graph is symmetric about y- ais; (ii) If f(-) = - f(), then the graph is symmetric about the origin Remark: If the graph is symmetric then only graph function for >0 and use symmetry to graph the corresponding part for <0 7. Make the sign chart for the reduced f() (i) plot -intercepts and points ecluded from the domain on the number line; these points divide the number line into a finite number of test intervals (ii) choose a point in each test interval and compute the value of f at the test point (iii) based on the sign of f at the test point, assign the sign to each test interval Remark: When f() > 0, then the graph of f is above the -ais. When f() < 0, then the graph is below the -ais 8. Sketch the graph of f using )-7): (i) Draw coordinate system and draw all asymptotes using a dashed line (ii) plot the intercepts, points where the graph crosses the horizontal/oblique asymptote and the points from the table in step 7. (iii) join the points with a continuous curve taking into consideration position of the graph relative to the -ais (step 7) and behavior near asymptotes.

19 Eample: Graph f ( ) 4 ) Domain: 4 = 0 = 4 =, = - Df = { -, } ) y intercept: y = f(0) = (-)/(-4)= 3 - intercepts: + = 0 (+4)(-3) = 0 = - 4 or = 3 ( 4)( 3) 3) Vertical asymptotes f ( ) 4 ( )( ) (-)(+) = 0 = = - vertical asymptotes: = -, = 4) Horizontal/oblique asymptotes Degree of numerator() = degree of the denominator(), y = /= is the horizontal asymptote 5) Intersection with asymptote: f() = The graph crosses the horizontal asymptote at = 8, that is at the point (8,) 6) Symmetries: f ( ) 4 7) ( ) f ( ) ( ) 4 4 f() is not the same as f(-), so f is not even and therefore not symmetric about y-ais ( ) f ( ) ( ) 4 4 f ( ) 4 4 f(-) and f() are not the same so, f is not odd and therefore not symmetric about the origin f ( ) 4-5 ( 5) ( 5) 8 positive ( 5) 4-3 ( 3) ( 3) 6 negative ( 3) positive

20 negative positive ) 5.4 Solving polynomial and rational inequalities Solving by graphing To solve an inequality f() > 0 (< 0, > 0, < 0) by graphing: (i) Graph function y = f(). Make sure to compute eactly the -intercepts (solve f() = 0) (ii) Read the solutions from the graph. To solve f() > 0, find the intervals (the values) on which the graph is above -ais but not on ais. To solve f() < 0, find the intervals (the values) on which the graph is below -ais, but not on the -ais. To solve f() > 0, find intervals (the values) on which the graph is above or on the -ais. To solve f() < 0, find intervals (the values) on which the graph is below or on the -ais.

21 Eample: The graph of a function f is given below. Use the graph to solve given inequality: a) f() > 0 for in (-,-) (-,) (4, +) f() > 0 for in (-,] [4, +) b) f() < 0 for in (-4,-3)(,3) f() < 0 for in [-4,-3) [, 3) 3 Eample: Solve ( ) ( 4) 0 3 Let f ( ) ( ) ( 4). f is a polynomial with zeros: 0,, -4. The multiplicities of zeros are, 3, respectively. The graph will cross the ais at 0 and and touch the -ais at -4. For large, f() behaves line y = 6. The graph of f() is below:

22 Now, f() > 0 when belongs to (-,-4) (-4,0) (, +). 3 Therefore the solution of ( ) ( 4) 0 is (-,-4) (-4,0) (, +). Algebraic methods Solving a Polynomial Inequality: polynomial > 0 (> 0, < 0, < 0) (i) Write the inequality in the standard form (0 on the right hand side) (ii) Find the zeros of the polynomial that is solve the equation : polynomial = 0 (iii) Plot the zeros on the number line (iv) The zeros divide the number line into a finite number of intervals on which the polynomial has the same sign. Choose a number in each interval (a test point) and evaluate the value of the polynomial at each number. (v) If the value of the polynomial at the chosen number is positive (> 0), then the polynomial is positive on the whole interval If the value of the polynomial at the chosen number is negative (< 0), then the polynomial is negative on the whole interval (vi) Choose, as the solution, the intervals on which the polynomial has a desired sign. Use interval notation. Include the endpoints only when the original inequality is < or >. If there are two separate intervals on which the polynomial has a desired sign, use the union sign,, between the intervals. Eample: Solve 3 > (i) 3 > 3 > 0 (ii) 3 = 0 ( - )= 0 = 0 or =0 = = (iii) (iv) interval Test point Value of 3 - at the test point (-, -) - (-) 3 (-) = -8 + = -6 negative (-,0) (-0.5) - (-0.5)= =.375 positive (0, ) 0.5 (.5) 3 -(.5)= negative (, ) 3 - = 6 positive

23 (v) (vi) Since the inequality is 3 > 0, we choose the intervals on which the polynomial is positive and include the endpoints. There are two intervals, so we use. Solution: [-,0] [,) P Solving a Rational Inequality: 0 Q ( 0, 0, 0), P, Q are polynomials (i) P Write the inequality in the standard form 0 Q ( 0, 0, 0) (0 on the right hand side) (ii) Solve the equations: P = 0 and Q = 0 (iii) Plot the solutions on the number line. Place open circle at each solution of Q = 0; those numbers cannot ever be included in the solution set (they make the denominator zero and are out of the domain) (iv) The solutions divide the number line into a finite number of intervals. A rational function will have a constant sign in each interval. Choose a number in each interval and evaluate the value of the epression P at each number. Q (v) P P If the value of is positive (> 0), then is positive on the whole interval Q Q If the value of Q P is negative (< 0), then Q P is negative on the whole interval (vi) Choose, as the solution, the intervals on which Q P has a desired sign. Use interval notation. Include the endpoints only when the original inequality is < or >. Remember to never include the endpoint with an open circle! If there are two or more such intervals, use the union sign,. Eample: Solve 4 (i) ( 4) (ii) Numerator = 0 denominator = 0 +0 = 0-4 = 0 = 0 = 4

24 (iii) (iv) interval Test point 0 Value of 4 (- 4) (4, 0) (0, +) at the test point negative positive negative (v) 0 (vi) Since the inequality is > 0, we choose the intervals on which 4 endpoints that do not have an open circle. Solution: (4, 0] 0 is positive and include 4

Power functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd

Power functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd 5.1 Polynomial Functions A polynomial unctions is a unction o the orm = a n n + a n-1 n-1 + + a 1 + a 0 Eample: = 3 3 + 5 - The domain o a polynomial unction is the set o all real numbers. The -intercepts

More information

Math Rational Functions

Math Rational Functions Rational Functions Math 3 Rational Functions A rational function is the algebraic equivalent of a rational number. Recall that a rational number is one that can be epressed as a ratio of integers: p/q.

More information

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

More information

Rational functions are defined for all values of x except those for which the denominator hx ( ) is equal to zero. 1 Function 5 Function

Rational functions are defined for all values of x except those for which the denominator hx ( ) is equal to zero. 1 Function 5 Function Section 4.6 Rational Functions and Their Graphs Definition Rational Function A rational function is a function of the form that h 0. f g h where g and h are polynomial functions such Objective : Finding

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important

More information

Section 4.4 Rational Functions and Their Graphs

Section 4.4 Rational Functions and Their Graphs Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, is a 16 rational function.

More information

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION 4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

Chapter 8. Examining Rational Functions. You see rational functions written, in general, in the form of a fraction: , where f and g are polynomials

Chapter 8. Examining Rational Functions. You see rational functions written, in general, in the form of a fraction: , where f and g are polynomials Chapter 8 Being Respectful of Rational Functions In This Chapter Investigating domains and related vertical asymptotes Looking at limits and horizontal asymptotes Removing discontinuities of rational functions

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest

More information

An Insight into Division Algorithm, Remainder and Factor Theorem

An Insight into Division Algorithm, Remainder and Factor Theorem An Insight into Division Algorithm, Remainder and Factor Theorem Division Algorithm Recall division of a positive integer by another positive integer For eample, 78 7, we get and remainder We confine the

More information

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS

CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided

More information

Rational Functions 5.2 & 5.3

Rational Functions 5.2 & 5.3 Math Precalculus Algebra Name Date Rational Function Rational Functions 5. & 5.3 g( ) A function is a rational function if f ( ), where g( ) and h( ) are polynomials. h( ) Vertical asymptotes occur at

More information

1.1 Solving a Linear Equation ax + b = 0

1.1 Solving a Linear Equation ax + b = 0 1.1 Solving a Linear Equation ax + b = 0 To solve an equation ax + b = 0 : (i) move b to the other side (subtract b from both sides) (ii) divide both sides by a Example: Solve x = 0 (i) x- = 0 x = (ii)

More information

LIMITS AND CONTINUITY

LIMITS AND CONTINUITY LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

More information

Algebra 1 Course Title

Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

More information

You are correct. The rational function becomes undefined when the denominator equals zero.

You are correct. The rational function becomes undefined when the denominator equals zero. 3.4. RATIONAL FUNCTIONS. A rational function is a function of the form where P() and Q() are polynomial functions and Q(). It is important be able identify the domain of rational functions. The domain

More information

Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4. _.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

More information

Rational Polynomial Functions

Rational Polynomial Functions Rational Polynomial Functions Rational Polynomial Functions and Their Domains Today we discuss rational polynomial functions. A function f(x) is a rational polynomial function if it is the quotient of

More information

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

More information

Name: where Nx ( ) and Dx ( ) are the numerator and

Name: where Nx ( ) and Dx ( ) are the numerator and Oblique and Non-linear Asymptote Activity Name: Prior Learning Reminder: Rational Functions In the past we discussed vertical and horizontal asymptotes of the graph of a rational function of the form m

More information

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) = Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could

More information

Contents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing...

Contents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing... Contents 6 Graph Sketching 87 6.1 Increasing Functions and Decreasing Functions.......................... 87 6.2 Intervals Monotonically Increasing or Decreasing....................... 88 6.3 Etrema Maima

More information

Class Notes for MATH 2 Precalculus. Fall Prepared by. Stephanie Sorenson

Class Notes for MATH 2 Precalculus. Fall Prepared by. Stephanie Sorenson Class Notes for MATH 2 Precalculus Fall 2012 Prepared by Stephanie Sorenson Table of Contents 1.2 Graphs of Equations... 1 1.4 Functions... 9 1.5 Analyzing Graphs of Functions... 14 1.6 A Library of Parent

More information

3.3. GRAPHS OF RATIONAL FUNCTIONS. Some of those sketching aids include: New sketching aids include:

3.3. GRAPHS OF RATIONAL FUNCTIONS. Some of those sketching aids include: New sketching aids include: 3.3. GRAPHS OF RATIONAL FUNCTIONS In a previous lesson you learned to sketch graphs by understanding what controls their behavior. Some of those sketching aids include: y-intercept (if any) x-intercept(s)

More information

3. Power of a Product: Separate letters, distribute to the exponents and the bases

3. Power of a Product: Separate letters, distribute to the exponents and the bases Chapter 5 : Polynomials and Polynomial Functions 5.1 Properties of Exponents Rules: 1. Product of Powers: Add the exponents, base stays the same 2. Power of Power: Multiply exponents, bases stay the same

More information

The Not-Formula Book for C1

The Not-Formula Book for C1 Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

More information

Polynomials Classwork

Polynomials Classwork Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th -degree polynomial function Def.: A polynomial function of degree n in the vaiable

More information

CLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column.

CLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column. SYNTHETIC DIVISION CLASS NOTES When factoring or evaluating polynomials we often find that it is convenient to divide a polynomial by a linear (first degree) binomial of the form x k where k is a real

More information

MTH 3005 - Calculus I Week 8: Limits at Infinity and Curve Sketching

MTH 3005 - Calculus I Week 8: Limits at Infinity and Curve Sketching MTH 35 - Calculus I Week 8: Limits at Infinity and Curve Sketching Adam Gilbert Northeastern University January 2, 24 Objectives. After reviewing these notes the successful student will be prepared to

More information

Solutions to Self-Test for Chapter 4 c4sts - p1

Solutions to Self-Test for Chapter 4 c4sts - p1 Solutions to Self-Test for Chapter 4 c4sts - p1 1. Graph a polynomial function. Label all intercepts and describe the end behavior. a. P(x) = x 4 2x 3 15x 2. (1) Domain = R, of course (since this is a

More information

McMurry University Pre-test Practice Exam. 1. Simplify each expression, and eliminate any negative exponent(s).

McMurry University Pre-test Practice Exam. 1. Simplify each expression, and eliminate any negative exponent(s). 1. Simplify each expression, and eliminate any negative exponent(s). a. b. c. 2. Simplify the expression. Assume that a and b denote any real numbers. (Assume that a denotes a positive number.) 3. Find

More information

3.5 Summary of Curve Sketching

3.5 Summary of Curve Sketching 3.5 Summary of Curve Sketching Follow these steps to sketch the curve. 1. Domain of f() 2. and y intercepts (a) -intercepts occur when f() = 0 (b) y-intercept occurs when = 0 3. Symmetry: Is it even or

More information

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

More information

Testing Center Student Success Center x x 18 12x I. Factoring and expanding polynomials

Testing Center Student Success Center x x 18 12x I. Factoring and expanding polynomials Testing Center Student Success Center Accuplacer Study Guide The following sample questions are similar to the format and content of questions on the Accuplacer College Level Math test. Reviewing these

More information

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real

More information

Lesson 6: Linear Functions and their Slope

Lesson 6: Linear Functions and their Slope Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation

More information

5.1 The Remainder and Factor Theorems; Synthetic Division

5.1 The Remainder and Factor Theorems; Synthetic Division 5.1 The Remainder and Factor Theorems; Synthetic Division In this section you will learn to: understand the definition of a zero of a polynomial function use long and synthetic division to divide polynomials

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

with "a", "b" and "c" representing real numbers, and "a" is not equal to zero.

with a, b and c representing real numbers, and a is not equal to zero. 3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,

More information

Polynomial Degree and Finite Differences

Polynomial Degree and Finite Differences CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

More information

In this lesson you will learn to find zeros of polynomial functions that are not factorable.

In this lesson you will learn to find zeros of polynomial functions that are not factorable. 2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

Section 5.0A Factoring Part 1

Section 5.0A Factoring Part 1 Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (

More information

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial

More information

Analyzing Polynomial and Rational Functions

Analyzing Polynomial and Rational Functions Analyzing Polynomial and Rational Functions Raja Almukahhal, (RajaA) Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as

More information

Algebra II Notes Rational Functions Unit Rational Functions. Math Background

Algebra II Notes Rational Functions Unit Rational Functions. Math Background Algebra II Notes Rational Functions Unit 6. 6.6 Rational Functions Math Background Previously, you Simplified linear, quadratic, radical and polynomial functions Performed arithmetic operations with linear,

More information

Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

More information

7.7 Solving Rational Equations

7.7 Solving Rational Equations Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

More information

ALGEBRA I A PLUS COURSE OUTLINE

ALGEBRA I A PLUS COURSE OUTLINE ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best

More information

Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions

Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions A) Vertical Asymptotes A rational function, in lowest terms, will have vertical asymptotes at the real zeros of the denominator

More information

Core Maths C1. Revision Notes

Core Maths C1. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

More information

Symmetry. A graph is symmetric with respect to the y-axis if, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

Symmetry. A graph is symmetric with respect to the y-axis if, for every point (x, y) on the graph, the point (-x, y) is also on the graph. Symmetry When we graphed y =, y = 2, y =, y = 3 3, y =, and y =, we mentioned some of the features of these members of the Library of Functions, the building blocks for much of the study of algebraic functions.

More information

Quadratic Equations and Inequalities

Quadratic Equations and Inequalities MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

More information

An inequality is a mathematical statement containing one of the symbols <, >, or.

An inequality is a mathematical statement containing one of the symbols <, >, or. Further Concepts for Advanced Mathematics - FP1 Unit 3 Graphs & Inequalities Section3c Inequalities Types of Inequality An inequality is a mathematical statement containing one of the symbols , or.

More information

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2 Pure Math 0 Notes Unit : Polynomials Unit : Polynomials -: Reviewing Polynomials Epressions: - mathematical sentences with no equal sign. Eample: Equations: - mathematical sentences that are equated with

More information

Precalculus A 2016 Graphs of Rational Functions

Precalculus A 2016 Graphs of Rational Functions 3-7 Precalculus A 2016 Graphs of Rational Functions Determine the equations of the vertical and horizontal asymptotes, if any, of each function. Graph each function with the asymptotes labeled. 1. ƒ(x)

More information

Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Method To Solve Linear, Polynomial, or Absolute Value Inequalities: Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

More information

Answers to Basic Algebra Review

Answers to Basic Algebra Review Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract

More information

Solutions for 2015 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama

Solutions for 2015 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama Solutions for 0 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama. For f(x = 6x 4(x +, find f(. Solution: The notation f( tells us to evaluate the

More information

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005 Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

More information

Objectives: To graph exponential functions and to analyze these graphs. None. The number A can be any real constant. ( A R)

Objectives: To graph exponential functions and to analyze these graphs. None. The number A can be any real constant. ( A R) CHAPTER 2 LESSON 2 Teacher s Guide Graphing the Eponential Function AW 2.6 MP 2.1 (p. 76) Objectives: To graph eponential functions and to analyze these graphs. Definition An eponential function is a function

More information

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if 1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c

More information

Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems

Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

3.6 The Real Zeros of a Polynomial Function

3.6 The Real Zeros of a Polynomial Function SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,

More information

Lecture 7 : Inequalities 2.5

Lecture 7 : Inequalities 2.5 3 Lecture 7 : Inequalities.5 Sometimes a problem may require us to find all numbers which satisfy an inequality. An inequality is written like an equation, except the equals sign is replaced by one of

More information

Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter 2 Limits Functions and Sequences sequence sequence Example Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

More information

Rational Functions ( )

Rational Functions ( ) Rational Functions A rational function is a function of the form r P Q where P and Q are polynomials. We assume that P() and Q() have no factors in common, and Q() is not the zero polynomial. The domain

More information

2-5 Rational Functions

2-5 Rational Functions -5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4

More information

Introduction to Finite Systems: Z 6 and Z 7

Introduction to Finite Systems: Z 6 and Z 7 Introduction to : Z 6 and Z 7 The main objective of this discussion is to learn more about solving linear and quadratic equations. The reader is no doubt familiar with techniques for solving these equations

More information

Methods to Solve Quadratic Equations

Methods to Solve Quadratic Equations Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a second-degree

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

More information

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , ) Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.1-1.4, 1.6-1.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

More information

Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

More information

ALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340

ALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340 ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:

More information

Able Enrichment Centre - Prep Level Curriculum

Able Enrichment Centre - Prep Level Curriculum Able Enrichment Centre - Prep Level Curriculum Unit 1: Number Systems Number Line Converting expanded form into standard form or vice versa. Define: Prime Number, Natural Number, Integer, Rational Number,

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

STRAND: ALGEBRA Unit 3 Solving Equations

STRAND: ALGEBRA Unit 3 Solving Equations CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

More information

Algebra II Semester Exam Review Sheet

Algebra II Semester Exam Review Sheet Name: Class: Date: ID: A Algebra II Semester Exam Review Sheet 1. Translate the point (2, 3) left 2 units and up 3 units. Give the coordinates of the translated point. 2. Use a table to translate the graph

More information

Higher Education Math Placement

Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

Official Math 112 Catalog Description

Official Math 112 Catalog Description Official Math 112 Catalog Description Topics include properties of functions and graphs, linear and quadratic equations, polynomial functions, exponential and logarithmic functions with applications. A

More information

Graphing Rational Functions

Graphing Rational Functions Graphing Rational Functions A rational function is defined here as a function that is equal to a ratio of two polynomials p(x)/q(x) such that the degree of q(x) is at least 1. Examples: is a rational function

More information

A Quick Algebra Review

A Quick Algebra Review 1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

More information

Mathematics Chapter 8 and 10 Test Summary 10M2

Mathematics Chapter 8 and 10 Test Summary 10M2 Quadratic expressions and equations Expressions such as x 2 + 3x, a 2 7 and 4t 2 9t + 5 are called quadratic expressions because the highest power of the variable is 2. The word comes from the Latin quadratus

More information

1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

More information

A.1 Radicals and Rational Exponents

A.1 Radicals and Rational Exponents APPENDIX A. Radicals and Rational Eponents 779 Appendies Overview This section contains a review of some basic algebraic skills. (You should read Section P. before reading this appendi.) Radical and rational

More information

Items related to expected use of graphing technology appear in bold italics.

Items related to expected use of graphing technology appear in bold italics. - 1 - Items related to expected use of graphing technology appear in bold italics. Investigating the Graphs of Polynomial Functions determine, through investigation, using graphing calculators or graphing

More information

CHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often

CHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often 7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.

More information

Section 3-3 Approximating Real Zeros of Polynomials

Section 3-3 Approximating Real Zeros of Polynomials - Approimating Real Zeros of Polynomials 9 Section - Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros

More information

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

More information

Introduction to Modular Arithmetic, the rings Z 6 and Z 7

Introduction to Modular Arithmetic, the rings Z 6 and Z 7 Introduction to Modular Arithmetic, the rings Z 6 and Z 7 The main objective of this discussion is to learn modular arithmetic. We do this by building two systems using modular arithmetic and then by solving

More information

3.4 Complex Zeros and the Fundamental Theorem of Algebra

3.4 Complex Zeros and the Fundamental Theorem of Algebra 86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

More information

The degree of a polynomial function is equal to the highest exponent found on the independent variables.

The degree of a polynomial function is equal to the highest exponent found on the independent variables. DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

More information

Review for Calculus Rational Functions, Logarithms & Exponentials

Review for Calculus Rational Functions, Logarithms & Exponentials Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information