Higher. Polynomials and Quadratics 64

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Higher. Polynomials and Quadratics 64"

Transcription

1 hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining the Equation of a Parabola 7 6 Solving Quadratic Inequalities 74 7 Intersections of Lines and Parabolas 76 8 Polnomials 77 9 Snthetic Division Finding Unknown Coefficients 8 11 Finding Intersections of Curves 84 1 Determining the Equation of a Curve Approimating Roots 88 HSN100 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still Notes. For more details about the copright on these notes, please see

2 Unit Polnomials and Quadratics OUTCOME 1 Polnomials and Quadratics 1 Quadratics A quadratic has the form numbers, provided a 0. You should alread be familiar with the following. a + b + c where a, b, and c are an real The graph of a quadratic is called a parabola. There are two possible shapes: concave up (if a > 0 ) concave down (if a < 0 ) This has a minimum turning point This has a maimum turning point To find the roots (i.e. solutions) of the quadratic equation we can use: factorisation; completing the square (see Section 3); the quadratic formula: EXAMPLES 1. Find the roots of ( + 1)( 3). Solve 3 a b c + +, b ± b = (this is not given in the eam). a or 3 = 1 = 3. ( )( ) or + 4 = 4 = 4. Page 64 HSN000

3 Unit Polnomials and Quadratics 3. Find the roots of We cannot factorise + 4 1, but we can use the quadratic formula: 4 ± ( 1) = 1 4 ± = 4 ± 0 = = ± = ± 5. Note If there are two distinct solutions, the curve intersects the -ais twice. If there is one repeated solution, the turning point lies on the -ais. If b < 0 when using the quadratic formula, there are no points where the curve intersects the -ais. Page 65 HSN000

4 Unit Polnomials and Quadratics The Discriminant Given a + b + c, we call b the discriminant. This is the part of the quadratic formula which determines the number of real roots of the equation a + b + c. If b If b > 0, the roots are real and unequal (distinct)., the roots are real and equal (i.e. a repeated root). If b < 0, the roots are not real; the parabola does not cross the -ais. EXAMPLE 1. Find the nature of the roots of a = 9 b = 4 c = 16 Since b b = = , the roots are real and equal.. Find the values of q such that Since a = 6 b = 1 c = q 6 1 q q = has real roots, b b q q q 4q 144 q = has real roots. 0 : two roots one root no real roots Page 66 HSN000

5 Unit Polnomials and Quadratics 3. Find the range of values of k for which the equation no real roots. k + 7 has For no real roots, we need a = k b = c = 7 b b < 0 : < 0 4 k ( 7) < k < 0 8k < 4 4 k < 8 k <. 4. Show that ( k ) ( k ) ( k ) has real roots for all real values of k. a = k + 4 b = 3k + c = k b 1 7 ( 3k ) 4( k 4)( k ) 9k 1k 4 ( k 4)( 4k 8) = + + = = k + k + k k = + + ( k ) 1k 36 = + 6. Since b ac ( k ) 4 = + 6 0, the roots are alwas real. 3 Completing the Square The process of writing a b c called completing the square. = + + in the form ( ) = a + p + q is Once in completed square form we can determine the turning point of an parabola, including those with no real roots. The ais of smmetr is = p and the turning point is ( p, q). + p = + p + p. For eample, The process relies on the fact that ( ) we can write the epression + 4 using the bracket ( + ) since when multiplied out this gives the terms we want with an etra constant term. Page 67 HSN000

6 Unit Polnomials and Quadratics This means we can rewrite the epression gives us the correct and terms, with an etra constant. + k using ( + k ) since this We will use this to help complete the square for = Step 1 Make sure the equation is in the form = a + b + c. Step Take out the -coefficient as a factor of the and terms. Step 3 Replace the + k epression and compensate for the etra constant. = ( ) = = 3( ( + ) 4) 3 = 3( + ) 1 3. Step 4 Collect together the constant terms. = 3( + ) 15. Now that we have completed the square, we can see that the parabola with equation = has turning point (, 15). EXAMPLES 1. Write = in the form = ( + p) + q. = = ( + 3) 9 5 = ( + 3) 14. Note You can alwas check our answer b epanding the brackets.. Write ( ) ( 3 ) in the form ( + p) + q. = = Page 68 HSN000

7 Unit Polnomials and Quadratics 3. Write = in the form = ( + a) + b and then state: (i) the ais of smmetr, and (ii) the minimum turning point of the parabola with this equation. = ( ) ( ) = = (i) The ais of smmetr is = 4. (ii) The minimum turning point is ( 4, 19). 4. A parabola has equation = (a) Epress the equation in the form = ( + a) + b. (b) State the turning point of the parabola and its nature. (a) = ( ) = (( 3 9 ) 4 ) ( 3 ) 3 ( ) = = = 4. (b) The turning point is 3 ( ), and is a minimum. Remember If the coefficient of is positive then the parabola is concave up. Page 69 HSN000

8 Unit Polnomials and Quadratics 4 Sketching Parabolas The method used to sketch the curve with equation depends on how man times the curve intersects the -ais. = a + b + c We have met curve sketching before. However, when sketching parabolas, we do not need to use calculus. We know there is onl one turning point, and we have methods for finding it. Parabolas with one or two roots Find the -ais intercepts b factorising or using the quadratic formula. Find the -ais intercept (i.e. where ). The turning point is on the ais of smmetr: O The ais of smmetr is halfwa between two distinct roots. O A repeated root lies on the ais of smmetr. Parabolas with no real roots There are no -ais intercepts. Find the -ais intercept (i.e. where ). Find the turning point b completing the square. EXAMPLES 1. Sketch the graph of = Since b 4 ac = ( 8 ) > 0, the parabola crosses the -ais twice. The -ais intercept ( ) : = ( 0) 8( 0) + 7 = 7 ( 0, 7 ). The -ais intercepts ( ) : ( 1)( 7) 1 or 7 = 1 = 7 ( 1, 0 ) ( 7, 0 ). The ais of smmetr lies halfwa between = 1 and = 7, i.e. = 4, so the -coordinate of the turning point is 4. Page 70 HSN000

9 Unit Polnomials and Quadratics We can now find the -coordinate: ( ) ( ) = = = 9. So the turning point is ( 4, 9).. Sketch the parabola with equation Since b 4ac ( 6) 4 ( 1) ( 9) 0 = 6 9. = =, there is a repeated root. The -ais intercept ( ) : = ( 0) 6( 0) 9 = 9 ( 0, 9 ). Since there is a repeated root, ( 3, 0) is the turning point. 3. Sketch the curve with equation The -ais intercept ( ) : = ( ) ( + 3)( + 3) + 3 = 3 = ( 3, 0 ). Since b 4 ac = ( 8 ) 4 13 < 0, there are no real roots. The -ais intercept ( ) : = ( 0) 8( 0) + 13 = 13 ( 0,13 ). So the turning point is (, 5 ). Complete the square: = O 3 9 = 6 9 ( ) = = ( ) O = ( ) + 5. = + = ( 4, 9) 8 13 O (, 5) Page 71 HSN000

10 Unit Polnomials and Quadratics 5 Determining the Equation of a Parabola Given the equation of a parabola, we have seen how to sketch its graph. We will now consider the opposite problem: finding an equation for a parabola based on information about its graph. We can find the equation given: the roots and another point, or the turning point and another point. When we know the roots If a parabola has roots where k is some constant. = a and = b then its equation is of the form = k ( a)( b) If we know another point on the parabola, then we can find the value of k. EXAMPLES 1. A parabola passes through the points ( 1, 0 ), ( 5, 0 ) and ( 0, 3 ). Find the equation of the parabola. Since the parabola cuts the -ais where = 1 and = 5, the equation is of the form: = k ( 1)( 5 ). To find k, we use the point ( 0, 3 ): = k ( 1)( 5) 3 = k ( 0 1)( 0 5) 3 = 5k k = 3 5. So the equation of the parabola is: = ( 1)( 5) ( 6 5) = = Page 7 HSN000

11 Unit Polnomials and Quadratics. Find the equation of the parabola shown below. ( 1, 6) Since there is a repeated root, the equation is of the form: = k ( + 5)( + 5) = k ( + 5 ) Hence = ( + ). 5 O To find k, we use ( 1, 6 ) : = k ( + 5) 6 = k ( 1+ 5) k = 6 =. When we know the turning point Recall from Completing the Square that a parabola with turning point p, q has an equation of the form ( ) where a is some constant. = a( + p) + q If we know another point on the parabola, then we can find the value of a. EXAMPLE 3. Find the equation of the parabola shown below. O ( 4, ) 7 Since the turning point is ( 4, ), the equation is of the form: = a ( 4). = Hence ( ) To find a, we use ( 0, 7) : ( ) = a 4 ( ) 7 = a a = 5 a = Page 73 HSN000

12 Unit Polnomials and Quadratics 6 Solving Quadratic Inequalities The most efficient wa of solving a quadratic inequalit is b making a rough sketch of the parabola. To do this we need to know: the shape concave up or concave down, the -ais intercepts. We can then solve the quadratic inequalit b inspection of the sketch. EXAMPLES 1. Solve + 1 < 0. The parabola with equation The -ais intercepts are given b: + 1 ( + 4)( 3) + 4 = 4 or 3 = 3. = + 1 is concave up. Make a sketch: 4 = So + 1 < 0 for 4 < < 3.. Find the values of for which The parabola with equation The -ais intercepts are given b: ( ) ( 3 + )( 3) 3 + or 3 = = 3. = is concave down. Make a sketch: = So for 3 3. Page 74 HSN000

13 Unit Polnomials and Quadratics 3. Solve > 0. The parabola with equation = is concave up. The -ais intercepts are given b: Make a sketch: ( 1)( 3) = or 3 = 1 = So > 0 for < 1 and > Find the range of values of for which the curve is strictl increasing. We have d 4 5 d = +. The curve is strictl increasing where > Make a sketch: ( 1)( + 5) 1 or + 5 = 1 = = Remember Strictl increasing means d 0 d >. = So the curve is strictl increasing for < 5 and > Find the values of q for which ( ) For no real roots, b < 0 : a = 1 b = q q b = q 4 = ( q 4)( q 4) q c = 1 q = q 8q + 16 q + q q has no real roots. ( ) ( )( ) q 10q 16. = + We now need to solve the inequalit q 10q + 16 < 0. The parabola with equation q q = is concave up. Page 75 HSN000

14 Unit Polnomials and Quadratics The -ais intercepts are given b: q 10q + 16 ( q )( q ) 8 q or q 8 Therefore b q = q = 8. < 0 for q 8 no real roots when < q < 8. Make a sketch: = q 10q + 16 < <, and so ( ) 7 Intersections of Lines and Parabolas 8 + q q has To determine how man times a line intersects a parabola, we substitute the equation of the line into the equation of the parabola. We can then use the discriminant, or factorisation, to find the number of intersections. q If b > 0, the line and curve intersect twice. If b, the line and curve intersect once (i.e. the line is a tangent to the curve). If b < 0, the line and the parabola do not intersect. EXAMPLES 1. Show that the line = 5 is a tangent to the parabola and find the point of contact. Substitute = 5 into: = + 5 = ( 1)( 1). Since there is a repeated root, the line is a tangent at = 1. = + To find the -coordinate, substitute = 1 into the equation of the line: = 5 1 = 3. So the point of contact is ( 1, 3 ). Page 76 HSN000

15 Unit Polnomials and Quadratics. Find the equation of the tangent to = + 1 that has gradient 3. The equation of the tangent is of the form = m + c, with m = 3, i.e. = 3 + c. Substitute this into c = c = Since the line is a tangent: b ( 3) 4 ( 1 c ) c 4c = 5 c = 5 4. Therefore the equation of the tangent is: = Note You could also do this question using methods from Differentiation. 8 Polnomials Polnomials are epressions with one or more terms added together, where each term has a number (called the coefficient) followed b a variable (such as ) raised to a whole number power. For eample: or The degree of the polnomial is the value of its highest power, for eample: has degree has degree 18. Note that quadratics are polnomials of degree two. Also, constants are 0 polnomials of degree zero (e.g. 6 is a polnomial, since 6 = 6 ). Page 77 HSN000

16 Unit Polnomials and Quadratics 9 Snthetic Division Snthetic division provides a quick wa of evaluating polnomials. 3 For eample, consider f ( ) = Evaluating directl, we find f ( 6) = 11. We can also evaluate this using snthetic division with detached coefficients. Step 1 Detach the coefficients, and write them across the top row of the table. Note that the must be in order of decreasing degree. If there is no term of a specific degree, then zero is its coefficient. Step Write the number for which ou want to evaluate the polnomial (the input number) to the left Step 3 Bring down the first coefficient Step 4 Multipl this b the input number, writing the result underneath the net coefficient. Step 5 Add the numbers in this column. Repeat Steps 4 and 5 until the last column has been completed. The number in the lower-right cell is the value of the polnomial for the input value, often referred to as the remainder = f ( 6) Page 78 HSN000

17 Unit Polnomials and Quadratics EXAMPLE 3 1. Given f ( ) = + 40, evaluate f ( ) using snthetic division So f ( ). using the above process Note In this eample, the remainder is zero, so f ( ). 3 This means + 40 when =, which means that = is a root of the equation. So + must be a factor of the cubic. We can use this to help with factorisation: ( ) f ( ) = ( + ) q ( ) where q ( ) is a quadratic Is it possible to find the quadratic q ( ) using the table? Tring the numbers from the bottom row as coefficients, we find: ( + )( 0) 3 = + 3 = 40 = f ( ) So using the numbers from the bottom row as coefficients has given the correct quadratic. In fact, this method alwas gives the correct quadratic, making snthetic division a useful tool for factorising polnomials. EXAMPLES. Show that 4 is a factor of 4 is a factor = 4 is a root Since the remainder is zero, = 4 is a root, so 4 is a factor. Page 79 HSN000

18 Unit Polnomials and Quadratics 3 3. Given f ( ) = , show that = 7 is a root of f ( ), and hence full factorise f ( ) Since the remainder is zero, = 7 is a root. 3 Hence we have f ( ) = Show that = 5 is a root of factorise the cubic ( 7)( 7 1) = + + Using snthetic division to factorise ( )( )( ) = =, and hence full Since = 5 is a root, + 5 is a factor. ( )( ) = This does not factorise an further since the quadratic has b < 0. In the eamples above, we have been given a root or factor to help factorise polnomials. However, we can still use snthetic division if we do not know a factor or root. Provided that the polnomial has an integer root, it will divide the constant term eactl. So b tring snthetic division with all divisors of the constant term, we will eventuall find the integer root. 5. Full factorise Numbers which divide 15: ± 1, ± 3, ± 5, ± Tr = 1: ( 1) + 5( 1) 8( 1) 15 = Tr = 1: ( 1) + 5( 1) 8( 1) 15 = Note For ±1, it is simpler just to evaluate the polnomial directl, to see if these values are roots. Page 80 HSN000

19 Unit Polnomials and Quadratics Tr = 3: Since = 3 is a root, 3 is a factor. 3 So = ( 3)( ) = ( 3)( + 1)( + 5 ). Using snthetic division to solve equations We can also use snthetic division to help solve equations. EXAMPLE 6. Find the solutions of =. Numbers which divide 1: ± 1, ±, ± 3, ± 4, ± 6, ± 1. 3 Tr = 1: ( 1) 15( 1) + 16( 1) + 1 = Tr = 1: ( 1) 15( 1) + 16( 1) + 1 Tr = : = Since = is a root, is a factor: = The Factor Theorem and Remainder Theorem For a polnomial f ( ): If f ( ) is divided b h or + 1 = 1 then the remainder is f ( ) f ( h) h is a factor of f ( ). h, and As we saw, snthetic division helps us to write f ( ) in the form where ( ) ( h) q ( ) + f ( h) q is called the quotient and f ( ) ( )( ) 11 6 ( )( + 1)( 6) h the remainder. or 6 = 6. Page 81 HSN000

20 Unit Polnomials and Quadratics EXAMPLE 3 7. Find the quotient and remainder when f ( ) = is divided b + 1, and epress ( ) The quotient is 4 3 ( ) ( )( ) f = f as ( 1) q( ) f ( h) and the remainder is 3, so 10 Finding Unknown Coefficients Consider a polnomial with some unknown coefficients, such as 3 + p p + 4, where p is a constant. If we divide the polnomial b h, then we will obtain an epression for the remainder in terms of the unknown constants. If we alread know the value of the remainder, we can solve for the unknown constants. EXAMPLES 1. Given that 3 is a factor of 3 is a factor = 3 is a root p p p p 3 p Since = 3 is a root, the remainder is zero: p 3 p = 4 p = , find the value of p. Note This is just the same snthetic division procedure we are used to. Page 8 HSN000

21 Unit Polnomials and Quadratics 3. When f ( ) = p + q q is divided b, the remainder is 6, and 1 is a factor of f ( ). Find the values of p and q. When f ( ) is divided b, the remainder is 6. p q 17 4q p 4 p + q 8 p + 4q 34 p p + q 4 p + q 17 8p + 8q 34 Since the remainder is 6, we have: 8 p + 8q 34 = 6 8 p + 8q = 40 p + q = 5. 1 Since 1 is a factor, f ( 1) : 3 f ( 1) = p( 1) + q ( 1) 17( 1) + 4q = p + q q = p + 5q 17 i.e. p + 5q = 17. Note There is no need to use snthetic division here, but ou could if ou wish. Solving 1 and simultaneousl, we obtain: 1: 4q = 1 q = 3. Put q = 3 into 1: p + 3 = 5 Hence p = and q = 3. p =. Page 83 HSN000

22 Unit Polnomials and Quadratics 11 Finding Intersections of Curves We have alread met intersections of lines and parabolas in this outcome, but we were mainl interested in finding equations of tangents We will now look at how to find the actual points of intersection and not just for lines and parabolas; the technique works for an polnomials. EXAMPLES 1. Find the points of intersection of the line = 4 4 and the parabola = 1. To find intersections, equate: 1 = ( )( ) = 1 or = 4. Find the -coordinates b putting the -values into one of the equations: when = 1, = 4 ( 1) 4 = 4 4 = 8, when = 4, = = 16 4 = 1. So the points of intersection are ( 1, 8) and ( 4,1 ).. Find the coordinates of the points of intersection of the cubic 3 = and the line = To find intersections, equate: = ( )( ) ( 1)( 3)( 5) = 1 or = 3 or = 5. Find the -coordinates b putting the -values into one of the equations: when = 1, = = =, when = 3, = = = 4, when = 5, = = = 10. So the points of intersection are ( ) 1,, ( 3, 4) and ( 5, 10). Remember You can use snthetic division to help with factorising. Page 84 HSN000

23 Unit Polnomials and Quadratics 3. The curves = + 4 and 3 = are shown below. 3 = A O B C = + 4 Find the -coordinates of A, B and C, where the curves intersect. To find intersections, equate: 3 + = + 3 ( )( ) ( )( )( ) = 1 or = or = 4. Remember You can use snthetic division to help with factorising. So at A, = 1; at B, = ; and at C, = Find the -coordinates of the points where the curves 3 and = intersect. = To find intersections, equate: = ( )( ) ( + 1)( 3)( 5) = 1 or = 3 or = 5. So the curves intersect where = 1,3,5. Page 85 HSN000

24 Unit Polnomials and Quadratics 1 Determining the Equation of a Curve Given the roots, and at least one other point ling on the curve, we can establish its equation using a process similar to that used when finding the equation of a parabola. EXAMPLE 1. Find the equation of the cubic shown in the diagram below. 6 3 O 1 36 Step 1 Write out the roots, then rearrange to get the factors. Step The equation then has these factors multiplied together with a constant, k. Step 3 Substitute the coordinates of a known point into this equation to find the value of k. Step 4 Replace k with this value in the equation. = 6 = 3 = = k ( + 6)( + 3)( 1 ). Using ( 0, 36) : k ( 0 + 6)( 0 + 3)( 0 1) = 36 k ( 3)( 1)( 6) = 36 18k = 36 = ( + 6)( + 3)( 1) k =. ( )( ) 3 ( ) = = = Page 86 HSN000

25 Unit Polnomials and Quadratics Repeated Roots If a repeated root eists, then a stationar point lies on the -ais. Recall that a repeated root eists when two roots, and hence two factors, are equal. EXAMPLE. Find the equation of the cubic shown in the diagram below. 9 O 3 = = 3 = So = k( + )( 3). Use ( 0, 9 ) to find k : 9 = k ( 0 + )( 0 3) 9 = k 9 k = 1. Note = 3 is a repeated root, so the factor ( 3) appears twice in the equation. 1 = = = So = ( + )( 3) ( )( 6 9) 3 ( ) 1 3 Page 87 HSN000

26 Unit Polnomials and Quadratics 13 Approimating Roots Polnomials have the special propert that if f ( a ) is positive and f ( b ) is negative then f must have a root between a and b. f ( a) a b f ( b) We can use this propert to find approimations for roots of polnomials to an degree of accurac b repeatedl zooming in on the root. EXAMPLE 3 Given f ( ) = 4 + 7, show that there is a real root between = 1 and =. Find this root correct to two decimal places. Evaluate f ( ) at = 1 and = : 3 f ( 1) = 1 4( 1) ( 1) + 7 = 3 f ( ) = 4( ) ( ) + 7 = 5 Since f ( 1) > 0 and f ( ) < 0, f ( ) has a root between these values. Start halfwa between = 1 and =, then take little steps to find a change in sign: f ( 1. 5) = < 0 f ( 1. 4) = < 0 f ( 1. 3) = < 0 f ( 1. ). 568 > 0. Since f ( 1. ) > 0 and f ( 1. 3) < 0, the root is between = 1. and = Start halfwa between = 1. and = 1. 3 : f ( 1. 5) > 0 f ( 1. 6) > 0 f ( 1. 7) > 0 f ( 1. 8) = < 0. Since f ( 1. 7) > 0 and f ( 1. 8) < 0, the root is between these values. Finall, f ( 1.75) > 0. Since f ( 1. 75) > 0 and f ( 1. 8) < 0, the root is between = and = Therefore the root is = 1. 8 to d.p. 1 Page 88 HSN000

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will

More information

SAMPLE. Polynomial functions

SAMPLE. Polynomial functions Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

More information

Core Maths C2. Revision Notes

Core Maths C2. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

More information

Polynomials Past Papers Unit 2 Outcome 1

Polynomials Past Papers Unit 2 Outcome 1 PSf Polnomials Past Papers Unit 2 utcome 1 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. Given p() = 2 + 6, which of the following are true? I. ( + 3) is a factor

More information

Chapter 6 Quadratic Functions

Chapter 6 Quadratic Functions Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where

More information

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

More information

When I was 3.1 POLYNOMIAL FUNCTIONS

When I was 3.1 POLYNOMIAL FUNCTIONS 146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

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

PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS

PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

Polynomials. Teachers Teaching with Technology. Scotland T 3. Teachers Teaching with Technology (Scotland)

Polynomials. Teachers Teaching with Technology. Scotland T 3. Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T Scotland Polynomials Teachers Teaching with Technology (Scotland) POLYNOMIALS Aim To demonstrate how the TI-8 can be used

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

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

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

More information

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t. REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

More information

Quadratic Equations and Functions

Quadratic Equations and Functions Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In

More information

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

More information

FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

More information

For each learner you will need: mini-whiteboard. For each small group of learners you will need: Card set A Factors; Card set B True/false.

For each learner you will need: mini-whiteboard. For each small group of learners you will need: Card set A Factors; Card set B True/false. Level A11 of challenge: D A11 Mathematical goals Starting points Materials required Time needed Factorising cubics To enable learners to: associate x-intercepts with finding values of x such that f (x)

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

Determinants can be used to solve a linear system of equations using Cramer s Rule.

Determinants can be used to solve a linear system of equations using Cramer s Rule. 2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

More information

Unit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials

Unit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial

More information

Identifying second degree equations

Identifying second degree equations Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

More information

Imagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x

Imagine a cube with any side length. Imagine increasing the height by 2 cm, the. Imagine a cube. x x OBJECTIVES Eplore functions defined b rddegree polnomials (cubic functions) Use graphs of polnomial equations to find the roots and write the equations in factored form Relate the graphs of polnomial equations

More information

FACTORING QUADRATICS 8.1.1 through 8.1.4

FACTORING QUADRATICS 8.1.1 through 8.1.4 Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten

More information

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking

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

CHAPTER 7: FACTORING POLYNOMIALS

CHAPTER 7: FACTORING POLYNOMIALS CHAPTER 7: FACTORING POLYNOMIALS FACTOR (noun) An of two or more quantities which form a product when multiplied together. 1 can be rewritten as 3*, where 3 and are FACTORS of 1. FACTOR (verb) - To factor

More information

3.2 The Factor Theorem and The Remainder Theorem

3.2 The Factor Theorem and The Remainder Theorem 3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial

More information

Click here for answers.

Click here for answers. CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent

More information

We start with the basic operations on polynomials, that is adding, subtracting, and multiplying.

We start with the basic operations on polynomials, that is adding, subtracting, and multiplying. R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

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

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

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Community College System Diagnostic and Placement Test Sample Questions North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

More information

2013 MBA Jump Start Program

2013 MBA Jump Start Program 2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of

More information

MATH 185 CHAPTER 2 REVIEW

MATH 185 CHAPTER 2 REVIEW NAME MATH 18 CHAPTER REVIEW Use the slope and -intercept to graph the linear function. 1. F() = 4 - - Objective: (.1) Graph a Linear Function Determine whether the given function is linear or nonlinear..

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

6706_PM10SB_C4_CO_pp192-193.qxd 5/8/09 9:53 AM Page 192 192 NEL

6706_PM10SB_C4_CO_pp192-193.qxd 5/8/09 9:53 AM Page 192 192 NEL 92 NEL Chapter 4 Factoring Algebraic Epressions GOALS You will be able to Determine the greatest common factor in an algebraic epression and use it to write the epression as a product Recognize different

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

Taylor Polynomials. for each dollar that you invest, giving you an 11% profit.

Taylor Polynomials. for each dollar that you invest, giving you an 11% profit. Taylor Polynomials Question A broker offers you bonds at 90% of their face value. When you cash them in later at their full face value, what percentage profit will you make? Answer The answer is not 0%.

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

More information

VISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University

VISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS

More information

UNCORRECTED PAGE PROOFS

UNCORRECTED PAGE PROOFS number and and algebra TopIC 17 Polynomials 17.1 Overview Why learn this? Just as number is learned in stages, so too are graphs. You have been building your knowledge of graphs and functions over time.

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

Sect. 1.3: Factoring

Sect. 1.3: Factoring Sect. 1.3: Factoring MAT 109, Fall 2015 Tuesday, 1 September 2015 Algebraic epression review Epanding algebraic epressions Distributive property a(b + c) = a b + a c (b + c) a = b a + c a Special epansion

More information

Understanding Basic Calculus

Understanding Basic Calculus Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

More information

CPM Educational Program

CPM Educational Program CPM Educational Program A California, Non-Profit Corporation Chris Mikles, National Director (888) 808-4276 e-mail: mikles @cpm.org CPM Courses and Their Core Threads Each course is built around a few

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

Chapter 8. Lines and Planes. By the end of this chapter, you will

Chapter 8. Lines and Planes. By the end of this chapter, you will Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes

More information

7.2 Quadratic Equations

7.2 Quadratic Equations 476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic

More information

CHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS

CHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS CHAPTER 0 SYSTEMS, MATRICES, AND DETERMINANTS PRE-CALCULUS: A TEACHING TEXTBOOK Lesson 64 Solving Sstems In this chapter, we re going to focus on sstems of equations. As ou ma remember from algebra, sstems

More information

Lecture 1 Introduction 1. 1.1 Rectangular Coordinate Systems... 1. 1.2 Vectors... 3. Lecture 2 Length, Dot Product, Cross Product 5. 2.1 Length...

Lecture 1 Introduction 1. 1.1 Rectangular Coordinate Systems... 1. 1.2 Vectors... 3. Lecture 2 Length, Dot Product, Cross Product 5. 2.1 Length... CONTENTS i Contents Lecture Introduction. Rectangular Coordinate Sstems..................... Vectors.................................. 3 Lecture Length, Dot Product, Cross Product 5. Length...................................

More information

2.1 Three Dimensional Curves and Surfaces

2.1 Three Dimensional Curves and Surfaces . Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

More information

4Unit 2 Quadratic, Polynomial, and Radical Functions

4Unit 2 Quadratic, Polynomial, and Radical Functions CHAPTER 4Unit 2 Quadratic, Polnomial, and Radical Functions Comple Numbers, p. 28 f(z) 5 z 2 c Quadratic Functions and Factoring Prerequisite Skills... 234 4. Graph Quadratic Functions in Standard Form...

More information

Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Copyrighted Material. Chapter 1 DEGREE OF A CURVE Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

More information

SECTION 5-1 Exponential Functions

SECTION 5-1 Exponential Functions 354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational

More information

Roots, Linear Factors, and Sign Charts review of background material for Math 163A (Barsamian)

Roots, Linear Factors, and Sign Charts review of background material for Math 163A (Barsamian) Roots, Linear Factors, and Sign Charts review of background material for Math 16A (Barsamian) Contents 1. Introduction 1. Roots 1. Linear Factors 4. Sign Charts 5 5. Eercises 8 1. Introduction The sign

More information

Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

More information

3.6. The factor theorem

3.6. The factor theorem 3.6. The factor theorem Example 1. At the right we have drawn the graph of the polynomial y = x 4 9x 3 + 8x 36x + 16. Your problem is to write the polynomial in factored form. Does the geometry of the

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

4 Constrained Optimization: The Method of Lagrange Multipliers. Chapter 7 Section 4 Constrained Optimization: The Method of Lagrange Multipliers 551

4 Constrained Optimization: The Method of Lagrange Multipliers. Chapter 7 Section 4 Constrained Optimization: The Method of Lagrange Multipliers 551 Chapter 7 Section 4 Constrained Optimization: The Method of Lagrange Multipliers 551 LEVEL CURVES 2 7 2 45. f(, ) ln 46. f(, ) 6 2 12 4 16 3 47. f(, ) 2 4 4 2 (11 18) 48. Sometimes ou can classif the critical

More information

t hours This is the distance in miles travelled in 2 hours when the speed is 70mph. = 22 yards per second. = 110 yards.

t hours This is the distance in miles travelled in 2 hours when the speed is 70mph. = 22 yards per second. = 110 yards. The area under a graph often gives useful information. Velocit-time graphs Constant velocit The sketch shows the velocit-time graph for a car that is travelling along a motorwa at a stead 7 mph. 7 The

More information

Find the Relationship: An Exercise in Graphing Analysis

Find the Relationship: An Exercise in Graphing Analysis Find the Relationship: An Eercise in Graphing Analsis Computer 5 In several laborator investigations ou do this ear, a primar purpose will be to find the mathematical relationship between two variables.

More information

The Factor Theorem and a corollary of the Fundamental Theorem of Algebra

The Factor Theorem and a corollary of the Fundamental Theorem of Algebra Math 421 Fall 2010 The Factor Theorem and a corollary of the Fundamental Theorem of Algebra 27 August 2010 Copyright 2006 2010 by Murray Eisenberg. All rights reserved. Prerequisites Mathematica Aside

More information

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

More information

Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto: explainwhycubicequationspossesseitheronerealrootorthreerealroots

Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto: explainwhycubicequationspossesseitheronerealrootorthreerealroots Cubic equations Acubicequationhastheform mc-ty-cubicequations-2009- ax 3 + bx 2 + cx + d = 0 where a 0 Allcubicequationshaveeitheronerealroot,orthreerealroots.Inthisunitweexplorewhythis isso. Thenwelookathowcubicequationscanbesolvedbyspottingfactorsandusingamethodcalled

More information

Five 5. Rational Expressions and Equations C H A P T E R

Five 5. Rational Expressions and Equations C H A P T E R Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.

More information

The Australian Curriculum Mathematics

The Australian Curriculum Mathematics The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year

More information

Using the Area Model to Teach Multiplying, Factoring and Division of Polynomials

Using the Area Model to Teach Multiplying, Factoring and Division of Polynomials visit us at www.cpm.org Using the Area Model to Teach Multiplying, Factoring and Division of Polynomials For more information about the materials presented, contact Chris Mikles mikles@cpm.org From CCA

More information

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

More information

Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

More information

COLLEGE ALGEBRA. Paul Dawkins

COLLEGE ALGEBRA. Paul Dawkins COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5

More information

1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives

1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of

More information

http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304

http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304 MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio

More information

Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

More information

Factorising quadratics

Factorising quadratics Factorising quadratics An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to

More information

Algebra 2 Unit 10 Tentative Syllabus Cubics & Factoring

Algebra 2 Unit 10 Tentative Syllabus Cubics & Factoring Name Algebra Unit 10 Tentative Sllabus Cubics & Factoring DATE CLASS ASSIGNMENT Tuesda Da 1: S.1 Eponent s P: -1, -7 Jan Wednesda Da : S.1 More Eponent s P: 9- Jan Thursda Da : Graphing the cubic parent

More information

Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.

Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method

More information

Vector Fields and Line Integrals

Vector Fields and Line Integrals Vector Fields and Line Integrals 1. Match the following vector fields on R 2 with their plots. (a) F (, ), 1. Solution. An vector, 1 points up, and the onl plot that matches this is (III). (b) F (, ) 1,.

More information

Algebra and Geometry Review (61 topics, no due date)

Algebra and Geometry Review (61 topics, no due date) Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

More information

f x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y

f x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y Fourier Series When the French mathematician Joseph Fourier (768 83) was tring to solve a problem in heat conduction, he needed to epress a function f as an infinite series of sine and cosine functions:

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

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

Simplification Problems to Prepare for Calculus

Simplification Problems to Prepare for Calculus Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.

More information

FACTORING POLYNOMIALS

FACTORING POLYNOMIALS 296 (5-40) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated

More information

Indiana University Purdue University Indianapolis. Marvin L. Bittinger. Indiana University Purdue University Indianapolis. Judith A.

Indiana University Purdue University Indianapolis. Marvin L. Bittinger. Indiana University Purdue University Indianapolis. Judith A. STUDENT S SOLUTIONS MANUAL JUDITH A. PENNA Indiana Universit Purdue Universit Indianapolis COLLEGE ALGEBRA: GRAPHS AND MODELS FIFTH EDITION Marvin L. Bittinger Indiana Universit Purdue Universit Indianapolis

More information

In this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).

In this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form). CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,

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

The Big Picture. Correlation. Scatter Plots. Data

The Big Picture. Correlation. Scatter Plots. Data The Big Picture Correlation Bret Hanlon and Bret Larget Department of Statistics Universit of Wisconsin Madison December 6, We have just completed a length series of lectures on ANOVA where we considered

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

The majority of college students hold credit cards. According to the Nellie May

The majority of college students hold credit cards. According to the Nellie May CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials

More information

The graphs of linear functions, quadratic functions,

The graphs of linear functions, quadratic functions, 1949_07_ch07_p561-599.qd 7/5/06 1:39 PM Page 561 7 Polynomial and Rational Functions 7.1 Polynomial Functions 7. Graphing Polynomial Functions 7.3 Comple Numbers 7.4 Graphing Rational Functions 7.5 Equations

More information

New Higher-Proposed Order-Combined Approach. Block 1. Lines 1.1 App. Vectors 1.4 EF. Quadratics 1.1 RC. Polynomials 1.1 RC

New Higher-Proposed Order-Combined Approach. Block 1. Lines 1.1 App. Vectors 1.4 EF. Quadratics 1.1 RC. Polynomials 1.1 RC New Higher-Proposed Order-Combined Approach Block 1 Lines 1.1 App Vectors 1.4 EF Quadratics 1.1 RC Polynomials 1.1 RC Differentiation-but not optimisation 1.3 RC Block 2 Functions and graphs 1.3 EF Logs

More information

Polynomial Operations and Factoring

Polynomial Operations and Factoring Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

More information

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

More information

SAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?

SAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3? SAT Math Hard Practice Quiz Numbers and Operations 5. How many integers between 10 and 500 begin and end in 3? 1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red

More information

M122 College Algebra Review for Final Exam

M122 College Algebra Review for Final Exam M122 College Algebra Review for Final Eam Revised Fall 2007 for College Algebra in Contet All answers should include our work (this could be a written eplanation of the result, a graph with the relevant

More information

Mathematics More Visual Using Algebra Tiles

Mathematics More Visual Using Algebra Tiles www.cpm.org Chris Mikles CPM Educational Program A California Non-profit Corporation 33 Noonan Drive Sacramento, CA 958 (888) 808-76 fa: (08) 777-8605 email: mikles@cpm.org An Eemplary Mathematics Program

More information