Further algebra. polynomial identities

Transcription

1 8 8A Polynomial identities 8B Partial fractions 8C Simultaneous equations areas of study The solution of simultaneous equations arising from the intersection of a line with a parabola, circle or rectangular hyperbola using algebra ebookplus 8a polynomial identities Digital doc 0 Quick Questions Before discussing the definition of a polynomial identity, it is important to remember some basic definitions. An algebraic epression is made up of terms. In the term a n, a is referred to as the coefficient of n. A constant is a term with no variable beside it. For eample is an algebraic epression made up of two terms. The coefficient of 3 is. The constant is 3. A polynomial identity is an identity of the form: k n n + k n - n - + k n - n k + k 0, n N where k n, k n -... are constants and n is an element of the set of natural numbers N. The degree of a polynomial is given by the highest value of n. Hence a polynomial of degree is linear, of degree is a quadratic, of degree 3 is a cubic, of degree is a quartic and so on. Worked eample Which of the following are polynomials? Give reasons for your answers. a b + c ( + 6) 5 a In order for to be a polynomial, the powers must all be greater than or equal to 0, which they are. The highest power of is 3. b In order for + to be a polynomial, the powers must all be greater than or equal to 0, which they are not. WriTe a is a polynomial of degree 3 since it has descending powers of and these powers are all greater than or equal to zero, i.e. n N. b This is not a polynomial since the second term has a power of -. maths Quest advanced General mathematics for the Casio Classpad

2 c In order for ( + 6) 5 to be a polynomial, the powers must all be greater than or equal to 0, which they are. The highest power of is 5. c This is a polynomial of degree 5, since when epanded, it has n N. Two polynomials are said to be equal if each -value generates the same y-value. Polynomials are identical if they are of the same degree and corresponding coefficients are equal. Therefore, if: a 3 + b + c + d then a, b 0, c - and d 8. If two polynomials are known to be equal, then the process of equating coefficients can be used to solve problems. Worked Eample If ( a + b) 3 - a -(b - c) +, then find the values of a, b and c. Method : Technology-free If (a + b) 3 - a - (b - c) +, then the each corresponding term must be equal. Equate the terms. Write 5 3 (a + b) 3 5 a + b [] - a - a - a [] - - (b - c) - - (b - c) b - c [3] Solve these equations using substitution. Substituting a - into equation [] gives ( - ) + b 5 b 9 Substituting b 9 into equation [3] gives 9 - c c 3 Write the answer. a -, b 9 and c Method : Technology-enabled On the Main screen, using the soft keyboard, tap: ) {N Enter the equations as shown. Then press E. Write the answer. a -, b 9 and c 3

3 Worked eample 3 Determine values of a and b if m + (m + am + )(m + bm + ). WriTe The right-hand side must first be epanded. m + m + bm 3 + m + am + abm + am + m + bm + m + (b + a)m 3 + ( + ab)m + (a + b)m + Equate the coefficients. The coefficients of m 3, m, and m are zero. 0m 3 (b + a)m 3 0 b + a [] 0m ( + ab)m 0 + ab [] 0m (a +b)m 0 a + b [3] 3 Solve for a and b. From equation [], b - a Substitute b - a into equation [] 0 - a a a ± a or a - and b - b Write the answer. When a, b - and when a -, b. Worked eample If - is a factor of , find the other factor. WriTe Method : Technology-free ebookplus Tutorial int-063 Worked eample Since the epression is cubic, the other factor must be a quadratic, hence it is of the form a + b + c ( - )(a + b + c) RHS a 3 + b + c - a - b - c a 3 + (b - a) + (c - b) - c Equate the coefficients. 3 a 3 a [] - 6 (b - a) - 6 b - a [] (c - b) c - b [3] 3 Solve for a, b and c. Substitute a into equation [] 6 b - a b - Substitute b - into equation [3] c - b c - 6 maths Quest advanced General mathematics for the Casio Classpad

4 Substitute the values for a, b and c into a + b + c and write the answer. Method : Technology-enabled On the Main screen, tap: Action Transformation factor Complete the entry line as: factor( ) Then press E. When a, b - and c - 6 then the quadratic factor of is Write the answer. The quadratic factor of is REMEMBER. A polynomial identity is an identity of the form k n n + k n - n - + k n - n k + k 0, n N where k n, k n -... are constants and n is an element of the set of natural numbers N.. The degree of a polynomial is given by the highest value of n. 3. Polynomials are identical if they are of the same degree and corresponding coefficients are equal.. If two polynomials are known to be equal, then the process of equating coefficients can be used to solve problems. Eercise 8A Polynomial identities WE For each of the following epressions: i state whether or not it is a polynomial ii if yes to i then give its degree. a + b c (3 + ) 3 d 3 + WE Find the values of a, b and c if (a + b) 3 + (b - c) + (a + c) Find the values of a, b and c if ( - )(a + b + c) + 6. Find constants a, b given that ( - ) ( + a) + b + c. 5 WE 3 Determine the values of a and b if m + 5 (m + am + 5)(m + bm + 5). 6 If a( + ) + b( + ) + c, find the values of a, b and c. If a 3 + b + c + d ( - ) (m + n), epress b in terms of c and d. 5

5 8 We If - is a factor of , find the other factor. 9 If + is a factor of , find the other factor. 8B 0 If + is a factor of , find the other factor. partial fractions When a function is epressed as one polynomial divided by another, f() g ( ), it is often desirable to epress this using partial fractions. h ( ) This enables the function to be graphed more easily and also helps with the process of integration (which you will learn about in Mathematical Methods CAS). proper fractions If g()and h() are both linear functions, then the function can be epressed as a proper fraction in the form: b f() A +. h ( ) ebookplus Interactivity int-095 Partial fractions Worked eample 5 Epress + 5 b in the form A Method : Technology-free Epress the numerator as ( - 3) + b; the value of b must be. Write the answer in the form b A Method : Technology-enabled On the Main screen, tap: Action Transformation propfrac Complete the entry line as: propfrac Then press E. WriTe + 5 ( - 3) Write the answer in the form b A maths Quest advanced General mathematics for the Casio Classpad

6 Consider the case where g() is a polynomial of degree and h() is a polynomial of degree. g ( ) In this case the function, f( ), is a proper fraction, since the numerator has a smaller h ( ) power than the denominator. For every linear factor (a + b) in the denominator, there will be a partial fraction of the form A f( ) a + b. For every repeated linear factor of the form (a + b) in the denominator, then the partial fractions will be of the form f( ) +. On occasions when it is impossible ( a + b) ( a + b) to epress the partial fractions in the form f( ) +, they can be written as ( a + b) ( a + b) C f( ) + + ( a + b) ( a + b) ( a + b). Worked Eample 6 Epress + 3 in partial fraction form Method : Technology-free Factorise the denominator The denominator has two linear factors so there will be two partial fractions of the form + ( - 8) ( + 5). 3 Epress the sum of the two fractions on the right as a single fraction. Write ( - 8)( + 5), R\{- 5, 8} ( - 8)( + 5) A ( + 5) + B ( - 8) ( - 8)( + 5) ( - 8)( + 5) Equate the numerators and simplify. + 3 A( + 5) + B( - 8) + 3 A + 5A + B - 8B + 3 (A + B) + 5A - 8B 5 Equate the coefficients to solve for A and B. (A + B) A + B - B A [] 6 Substitute the values for A and B and write the answer in the form +. ( - 8) ( + 5) 3 5A - 8B [] Substitute equation [] into equation []. 3 5( - B) - 8B 3B B 3 A ( - 8) 3( + 5), R\{ - 5, 8}

7 Method : Technology-enabled On the Main screen, complete the entry line as: Highlight the equation and tap: Interactive Transformation epand Partial Fraction OK Write the answer in the form + ( - 8) ( + 5) ( - 8) 3( + 5), R \ { - 5, 8} Worked eample Epress - ( - )( + ) in partial fractions. WriTe ebookplus Tutorial int-06 Worked eample Method : Technology-free The denominator has one linear factor and one repeated linear factor so there will be three partial fractions of the form C + + ( - ) ( + ) ( + ). Epress the sum of the three fractions on the right as a single fraction. - C + + ( - )( + ) ( - ) ( + ) ( + ), R\{ -, }. - ( - )( + ) A ( + ) + B ( - )( + ) + C( -) ( - )( + ) 3 Equate the numerators and simplify. - A( + + ) + B ( - - ) + C ( - ) - A + A + A + B - B - B + C - C - (A + B) + (A - B + C) + A - B - C Equate the coefficients to solve for A, B and C. 0 (A + B) 0 A + B A - B [] (A - B + C) A - B + C [] - A - B - C [3] 8 maths Quest advanced General mathematics for the Casio Classpad

8 5 Substitute equation [] into equations [] and [3]. - 3B + C - 3B - C - 6 Solve these equations simultaneously. 3C 3 C Hence, - 3B + - 3B B 3 A 3 Substitute the values for A, B and C and write the answer in the form C + + ( - ) ( + ) ( + ). Method : Technology-enabled On the Main screen, complete the entry line as: - ( - )( + ) Highlight the equation and tap: Interactive Transformation epand Partial Fraction OK ( - )( + ) 3( - ) 3( + ) ( + ), R\{ -, } Write the answer in the form C + + ( - ) ( + ) ( + ) ( - )( + ) 3( - ) 3( + ) ( + ), R\{ -, } Sometimes the denominator may consist of an irreducible quadratic (a quadratic which cannot be factorised using real numbers). These types of functions need to be epressed in partial fractions of the form: C f( ) a + b + +. c + d + e Worked Eample 8 Epress in partial fractions. 3-8 Method : Technology-free Factorise the denominator. Write ( - )( + + ) 9

9 The denominator has a linear factor and an irreducible quadratic factor so the partial fractions will be of the form C Epress the sum of the two fractions on the right as a single fraction C R\{} A ( + + ) + ( B+ C)( - ) ( - )( + + ) Equate the numerators and simplify A( + + ) + (B + C) ( - ) A + A + A + B - B + C - C) (A + B) + (A - B + C) + A -C 5 Equate the coefficients to solve for A, B and C. 5 (A + B) A + B 5 B 5 - A [] 9 (A - B + C) 9 A - B + C [] 6 Substitute equation [] into equation [] and then subtract equation [3] to solve for A, B and C. Substitute the values for A, B and C and write the answer in the form C Method : Technology-enabled On the Main screen, complete the entry line as: Highlight the equation and tap: Interactive Transformation epand Partial Fraction OK 0 A - C [3] Substituting [] into [] A - (5 - A) + C 9 A C 9 A + C 9 Subtracting equation [3] 3C 9 C 3 A B , R\{} Maths Quest Advanced General Mathematics for the Casio ClassPad

10 Write the answer in the form C , R\{} + + improper fractions g ( ) In the case where g() has a higher power than h() the function f( ) is an improper h ( ) fraction. In this case, division of polynomials needs to be performed first either by long division or synthetic division. Worked eample 9 Epress as a partial fraction. - WriTe ebookplus Tutorial int-065 Worked eample 9 Method : Technology-free The degree of the denominator is less than the degree of the numerator, so division must be performed first. Divide the numerator by the denominator using long division. 3 Epress the answer as partial fractions. - is the divisor ) , R\{} - - Method : Technology-enabled On the Main screen, complete the entry line as: Highlight the equation and tap: Interactive Transformation epand Partial Fraction OK Write the answer , R\{} - - 8

11 remember g ( ) For rational functions of the form f( ) : h ( ) If g () and h () are both linear functions, then the function can be epressed in the b form f( ) A +. h ( ) Where the numerator is a linear function and the denominator is a quadratic which can be factorised, then the partial fraction will be of the form f( ) a + b + c + d. When the denominator has repeated linear factors of the form (a + b) then the partial fractions will be of the form f( ) + ( a + b) ( a + b). On occasions when it is impossible to epress the partial fractions in the form f( ) +, they can be written as ( a + b) ( a + b) C f( ) + + ( a + b) ( a + b) ( a + b). When the denominator contains an irreducible quadratic then the partial fractions will C be of the form f( ) a + b + +. c + d + e In the case where g () has a higher power than h () the function is an improper fraction so division of polynomials needs to be performed either by long division or synthetic division. eercise 8B ebookplus Digital doc WorkSHEET 8. partial fractions We5 Epress each of the following as the sum of two terms. a - 3 b + + c We6 Epress each of the following as partial fractions. + 6 a b ( + )( -) - 5 c We Epress each of the following as partial fractions. a - ( + ) b - c ( - )( + 3) We8 Epress each of the following as partial fractions a b c ( + )( - + 5) ( ) ( )( - ) 5 We9 Epress each of the following as partial fractions. a b c d d d ( - )( + ) d d maths Quest advanced General mathematics for the Casio Classpad

12 8c Simultaneous equations It is impossible to solve one linear equation with two unknown variables. There must be two equations with the same two unknowns for a solution to be found. Such equations are called simultaneous equations. There are several different ways to solve simultaneous equations. In this section we consider algebraic solutions of simultaneous equations arising from the intersection of a line with a parabola, circle or rectangular hyperbola. Worked Eample 0 Solve simultaneously: y and y Write Method : Technology-free Write the equations and label them [] and []. y [] y [] Substitute equation [] into equation []. Substituting [] into []: Transpose to make the RHS equal 0 and simplify Factorise. ( + ) 0 5 Solve for Substitute instead of into equation []. Substituting into []: y Write the answer. Solution set: (, ) Method : Technology-enabled On the Main screen, using the soft keyboard, tap: ) {N Enter the equations as shown. Then press E. Write the answer. Solving y and y for and y gives - and y -. That is, ( -, - ). 83

13 Worked eample Solve simultaneously: y + and + y. Method : Technology-free Write the equations and label them [] and []. WriTe ebookplus Tutorial int-066 Worked eample y + [] + y [] Substitute equation [] into equation []. Substituting [] into []: + ( + ) 3 Epand ( + ), using the perfect square identity and transpose to make the RHS Solve for, using the quadratic formula. a, b, c ± - (-3) ± + ± 8 ± - ± + 5 Write the two values of separately., Substitute - + instead of into equation [] and simplify. Substitute - - instead of into equation [] and simplify. -+ y y maths Quest advanced General mathematics for the Casio Classpad

14 8 Write the answer. (Make sure the values of and y are matched properly; that is, is placed with y and with y.) Solution set:,, - - -, Method : Technology-enabled On the Main screen, using the soft keyboard, tap: ) {N Enter the equations as shown. Then press E. Write the answer. Solving y + and + y for and y gives - ( + ) and y That is, and y - ( -) or + ( + ) ( -), or - +,. Worked Eample Solve simultaneously: y and y - 3. Method : Technology-free Write the equations and label them [] and []. Write Substitute equation [] into equation []. Substituting [] into []: y - [] y - 3 []

15 3 Solve for : (a) Multiply both sides of the equation by ( 3). (b) Epand and make the RHS 0. (c) Identify the values of a, b and c. (d) Substitute the values of a, b and c into the quadratic formula and simplify. ( )( 3) + 0 a, b, c ± (-) - ± 9-8 ± Write the two values of separately. +, - 5 Substitute + + into [] and simplify. y - 6 Substitute into [] and simplify. y Write the answer (leave it in surd form). Solution set:,, Method : Technology-enabled - 5 -, On the Main screen, using the soft keyboard, tap: ) {N Enter the equations as shown. Then press E. 86 Maths Quest Advanced General Mathematics for the Casio ClassPad

16 Write the answer. Solving y - and y for and y gives ( - ) y - ( - 5) and or and y That is, - ( - - ) ( - 5), or + + 5,. remember Simultaneous equations, arising from the intersection of a line with a parabola, circle or a rectangular hyperbola, can be solved using algebra as follows:. Transpose one of the equations (it is better to choose a linear equation) to make either or y the subject and substitute into the other equation.. Simplify the resulting equation (if properly simplified, it will result in a quadratic equation). 3. Solve the quadratic equation to find the value(s) of one variable.. Substitute the value(s) of the first variable into either of the two equations (preferably into the transposed one) and solve for the second variable. 5. Write the solution set. eercise 8C simultaneous equations We 0 Solve each of the following simultaneously. a y, y b y -, y c y, y + + d y 3, y e y -, y - - f y + 5, y + g y +, y h + 3y 6, y - i - y, y j y + 6, y - 3 We Solve each of the following simultaneously. a y, y + b y -, y + c y, + y d y -, y + e + y, + y f y , ( + 3) + y 6 g y -, ( - ) + y - 0 h y + 3, ( - ) + (y + 3) ebookplus Digital doc SkillSHEET 8. Using substitution to solve simultaneous equations i + y, + (y + ) 5 j 6-3y, ( - ) + (y - ) We Solve each of the following simultaneously. a y, y b y +, y - c y 3 -, y d y -, y

17 ebookplus Digital doc WorkSHEET 8. e y - 6, y f + y - 8 0, y + - h y -, y 0 g - 3 y, y - i - 3y, y j + y 5, - 3 y + mc Which of the following represent the solution to the pair of simultaneous equations + y 6 and y ? i - ( 6 3 3), ii ( -, ) iii ( -, ) iv ( -, 0) A i only B i and ii C ii and iv D ii and iii E i and iv 5 Buttons are to be attached to a shirt as shown on the diagram at right. If we draw a set of aes through the centre of the button, the position of the two holes can be described as the points of intersection of the line y with the circle + y. The other two holes are positioned at the points of intersection of the line y - with the same circle. Find the coordinates of the four holes. Give the answer correct to decimal places. y y + y 88 maths Quest advanced General mathematics for the Casio Classpad

18 Summary Polynomials A polynomial identity is an identity of the form: k n n + k n - n - + k n - n k + k 0, n N where k n, k n -... are constants and n is contained within the set of natural numbers N. The degree of a polynomial is given by the highest value of n. Polynomials are identical if they are of the same degree and corresponding coefficients are equal. If two polynomials are known to be equal, then the process of equating coefficients can be used. Partial fractions g ( ) For rational functions of the form f( ) : h ( ) b If g() and h() are both linear functions, then the function can be epressed in the form f( ) A +. h ( ) Where the numerator is a linear function and the denominator is a quadratic which can be factorised, then the partial fraction will be of the form f( ) a + b + c + d. When the denominator has repeated linear factors of the form ( a + b) then the partial fractions will be of the form f( ) +. On occasions when it is impossible to epress the partial fractions in ( a + b) ( a + b) the form f( ) + ( a + b) ( a + b), they can be written as f C ( ) + + ( a + b) ( a + b) ( a + b). When the denominator contains an irreducible quadratic then the partial fractions will be of the form + C f( ) +. ( a + b) c + d + e In the case where g() has a higher power than h(), the function is then an improper fraction so division of polynomials needs to be performed either by long division or synthetic division. Simultaneous equations Simultaneous equations, arising from the intersection of a line with a parabola, circle or a rectangular hyperbola, can be solved using algebra as follows: Transpose one of the equations (it is better to choose a linear equation) to make either or y the subject and substitute into the other equation. Simplify the resulting equation (if properly simplified, it will result in a quadratic equation). Solve the quadratic equation to find the value(s) of one variable. Substitute the value(s) of the first variable into either of the two equations (preferably into the transposed one) and solve for the second variable. Write the solution set. 89

19 chapter review Short answer Determine the values of a and b where ( + a + )( + b - ). - is a factor of ; find the other factor. - 3 Epress as partial fractions Epress as partial fractions. - 5 Find the coordinates of the points of intersection of the line y 5 with the hyperbola y Find the coordinates of the points of intersection of the line y with the parabola y -. Multiple choice If ( - )(a + b + c), then the values of a, b and c are: A a, b, c 6 B a -, b -, c 6 C a, b, c - 6 D a, b -, c 6 E a, b, c - 5 If a( - b)( - c) then the values of a, b and c are A a, b 3, c B a, b 3, c C a, b 3, c E a, b 3, c etended response D a, b 3, c If +, then: ( + )( - ) + - A A, B 3 B A 3, B C A, B D A, B E A, B 3 If C +, then the values of A, B and C would be: A A, B, C 3 B A, B 3, C C A, B, C 5 D A - 3, B, C 5 E A 3, B, C A solution to the pair of simultaneous equations 3 y.5 and y +. is: 0( - ) A D 5 3, 3 9, B 5 9, E none of these C 3 5, 6 The equation y - and (y ) + ( 3) 9 are solved simultaneously. When one of the equations is substituted into the other and the resultant equation is transposed to the form a + b + c 0, the values of a, b and c are: A, - 6, 8 B 6, - 6, 0 C, - 6, D 6, - 6, 9 E 6, - 6, 9 Find the coordinates of the points of intersection of the line y with: a the hyperbola y b the circle (y + ) + 0. In each case give the answer correct to decimal places. Consider the design, shown on the diagram at right: If we take the point of intersection of the straight lines to be an origin, the design can be described by the following system of equations: y y y y - y y B A T C R D S E G F Q P O H I M N J K L 90 Maths Quest Advanced General Mathematics for the Casio ClassPad

20 As can be seen from the diagram, there are 0 points of intersection (not counting the centre point). a What is the radius of a circle described by the equation + y? b Using the answer to a, state the coordinates of points A, F, K and P. c Find the coordinates of point I by solving an appropriate pair of simultaneous equations algebraically. Leave the answer in a surd form. d State the coordinates of points C, R and N using symmetry. e Find the coordinates of point H by solving algebraically an appropriate pair of simultaneous equations. f Using the symmetry of the design and your answer to part e, write the coordinates of points D, S and M. g State the points of intersection of the hyperbola y and the circle. h Find the coordinates of the points in question g by solving an appropriate pair of equations graphically, using a table of values or one of the iteration methods. Give the answer correct to decimal places. i State the points of intersection of the hyperbola y - and the circle. j Choose a method and use it to find the coordinates of the points in question i. Give the answer correct to decimal places. 3 A section of a roller coaster track is shown at right. It consists of three parts with the following equations: AB: h - 3 d BC: h d - 3d + 6 CDE: h 0.0d 3 -.5d + 5d -.56 where h is the height of the track above the ground level and d is the horizontal distance from A. a Find the coordinates of point B, by solving a pair of simultaneous equations algebraically. b The track is closest to the ground when it is 8 metres horizontally from A. What is its height at that point? c Find the horizontal distance(s) from A, when the car is 6 metres above ground level. d Use a CAS calculator to find the coordinates of point C. e By using a table of values or otherwise, find the h coordinates of point D. E f Point E is 30 metres horizontally from A and is the D highest point of this section of the track. Find the maimum height of the track. A g The track runs alongside the amusements pavilion. The roof of the pavilion follows the rule h 0.d +. As seen from the diagram, the car, while on this section of the B C track, will be level with the roof four times. Find the height of the car above the ground at each of these four points. Distance d ebookplus Height (m) Digital doc Test Yourself 9

21 ebookplus activities Chapter opener Digital doc 0 Quick Questions: Warm up with ten quick questions on further algebra. (page ) 8A Polynomial identities Tutorial We int-063: Watch how to find the quadratic factor of a cubic given the linear factor. (page ) 8B Partial fractions Tutorials We int-06: Watch how to epress a linear function divided by a cubic as a partial fraction. (page 8) We9 int-065: Watch how to epress a cubic divided by a linear function as a partial fraction. (page 8) Digital doc WorkSHEET 8.: Use the bisection and secant methods as well as the null factor law to solve simple and comple simultaneous equations, and apply learning to worded problems. (page 8) Interactivity Partial fractions int-095: Consolidate your understanding of how to determine partial fractions. (page 6) 8C Simultaneous equations Tutorial We int-066: Watch how to solve simultaneously a linear and an elliptical equation. (page 8) Digital docs SkillSHEET 8.: Practise using substitution to solve simultaneous equations. (page 8) WorkSHEET 8.: Practise finding solutions to linear and non-linear simultaneous equations. (page 88) Chapter review Digital doc Test Yourself: Take the end-of-chapter test to test your progress. (page 9) To access ebookplus activities, log on to 9 maths Quest advanced General mathematics for the Casio Classpad