Higher. Polynomials and Quadratics 64


 Amie Johnson
 4 years ago
 Views:
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 lowerright 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
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 informationSAMPLE. 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 informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More informationCore 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 informationPolynomials 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 informationChapter 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 information3.3. 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 informationPOLYNOMIAL 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 informationWhen 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 informationZero: 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 informationImplicit Differentiation
Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some
More informationPolynomial 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 informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More information10.1. 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 informationZeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.
_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms oneone and manone mappings understand the terms domain and range for a mapping understand the
More informationAssessment Schedule 2013
NCEA Level Mathematics (9161) 013 page 1 of 5 Assessment Schedule 013 Mathematics with Statistics: Apply algebraic methods in solving problems (9161) Evidence Statement ONE Expected Coverage Merit Excellence
More informationPROPERTIES 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 informationTo Be or Not To Be a Linear Equation: That Is the Question
To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not
More informationLesson 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 yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationINVESTIGATIONS 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 informationPolynomials. 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 TI8 can be used
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More information15.1. Exact Differential Equations. Exact FirstOrder Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact FirstOrder Equations 09 SECTION 5. Eact FirstOrder Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationPolynomial 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 informationJUST 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 informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph firstdegree equations. Similar methods will allow ou to graph quadratic equations
More informationCore 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 informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationSTRAND: 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 informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radioactive substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radioactive 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 information4.9 Graph and Solve Quadratic
4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationQuadratic 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 informationDownloaded 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 informationChapter 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 informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationFor each learner you will need: miniwhiteboard. 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 xintercepts with finding values of x such that f (x)
More informationFACTORING 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 informationSection 33 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 informationYear 9 set 1 Mathematics notes, to accompany the 9H book.
Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + 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 informationPolynomials. 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 informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationDeterminants 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 information2.3 Quadratic Functions
88 Linear and Quadratic Functions. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions:
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationIdentifying 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 informationI think that starting
. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries
More informationComplex Numbers. (x 1) (4x 8) n 2 4 x 1 2 23 No realnumber solutions. From the definition, it follows that i 2 1.
7_Ch09_online 7// 0:7 AM Page 99. Comple Numbers 9 SECTION 9. OBJECTIVES Epress square roots of negative numbers in terms of i. Write comple numbers in a bi form. Add and subtract comple numbers. Multipl
More information7.3 Parabolas. 7.3 Parabolas 505
7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of
More informationSolving Systems of Equations
Solving Sstems of Equations When we have or more equations and or more unknowns, we use a sstem of equations to find the solution. Definition: A solution of a sstem of equations is an ordered pair that
More informationMathematical goals. Starting points. Materials required. Time needed
Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between
More informationUnit 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 informationImagine 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 information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle  in particular
More informationSystems of Linear Equations: Solving by Substitution
8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing
More informationSystems of Equations Involving Circles and Lines
Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system
More informationFACTORING 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 informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationStudents 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 informationVersion 1.0. General Certificate of Education (Alevel) January 2012. Mathematics MPC4. (Specification 6360) Pure Core 4. Final.
Version.0 General Certificate of Education (Alevel) January 0 Mathematics MPC (Specification 660) Pure Core Final Mark Scheme Mark schemes are prepared by the Principal Eaminer and considered, together
More informationCHAPTER 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 informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationMathematics 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 information2.5 Zeros of a Polynomial Functions
.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the xaxis and
More information3.2 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 informationIntegrating algebraic fractions
Integrating algebraic fractions Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate
More informationWe 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 informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More information1 Maximizing pro ts when marginal costs are increasing
BEE12 Basic Mathematical Economics Week 1, Lecture Tuesda 12.1. Pro t maimization 1 Maimizing pro ts when marginal costs are increasing We consider in this section a rm in a perfectl competitive market
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle  in particular the basic equation representing
More informationClick 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 informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More information7.7 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 information5.2 Inverse Functions
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
More informationTHE PARABOLA 13.2. section
698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.
More informationReview 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 informationSECTION 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 informationSection 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 information2013 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 informationEQUATIONS OF LINES IN SLOPE INTERCEPT AND STANDARD FORM
. Equations of Lines in SlopeIntercept and Standard Form ( ) 8 In this SlopeIntercept Form Standard Form section Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications (0,
More informationNorth 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 informationFunctions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study
Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 21 Functions 22 Elementar Functions: Graphs and Transformations 23 Quadratic
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More information6 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 informationAlgebra 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{ } Sec 3.1 Systems of Linear Equations in Two Variables
Sec.1 Sstems of Linear Equations in Two Variables Learning Objectives: 1. Deciding whether an ordered pair is a solution.. Solve a sstem of linear equations using the graphing, substitution, and elimination
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More informationMATH 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 informationALGEBRA 1 SKILL BUILDERS
ALGEBRA 1 SKILL BUILDERS (Etra Practice) Introduction to Students and Their Teachers Learning is an individual endeavor. Some ideas come easil; others take timesometimes lots of time to grasp. In addition,
More informationWarmUp y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.
CST/CAHSEE: WarmUp Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More information6706_PM10SB_C4_CO_pp192193.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 information135 Final Review. Determine whether the graph is symmetric with respect to the xaxis, the yaxis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, 6); P2 = (7, 2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the ais, the ais, and/or the
More information