A4 - POLYNOMIALS AND PARTIAL FRACTIONS

A4: POLYNOMIALS AND PARTIAL FRACTIONS ● Multiplication and division of polynomials ● Use of remainder and factor theorems, including factorising polynomials and solving cubic equations ● Use of: - 𝑎 3 + 𝑏 3 = ( 𝑎 + 𝑏 ) 𝑎 2 − 𝑎𝑏 + 𝑏 2 ( )- 𝑎 3 − 𝑏 3 = ( 𝑎 − 𝑏 ) 𝑎 2 + 𝑎𝑏 + 𝑏 2 ( )● Partial fractions with cases where the denominator is no more complicated than: - ( 𝑎𝑥 + 𝑏 ) ( 𝑐𝑥 + 𝑑 )- ( 𝑎𝑥 + 𝑏 ) ( 𝑐𝑥 + 𝑑 ) 2 - ( 𝑎𝑥 + 𝑏 ) 𝑥 2 + 𝑐 2 ( ) 1. Express in partial fractions. 2 + 𝑥 2 𝑥 + 2 ( ) 2 𝑥 + 4 ( ) [4] 2. The function is defined by for all 𝑓 ( 𝑥 ) 𝑓 ( 𝑥 ) = 3 𝑥 2 + ℎ 𝑥 2 + 𝑘𝑥 − 4 real values of . Given that is a factor of and that when 𝑥 3 𝑥 − 1 𝑓 ( 𝑥 ) is divided by , the remainder is , find the value of each 𝑓 ( 𝑥 ) 𝑥 + 1 − 4 of the constants and . ℎ 𝑘 [5] 3. (a) Factorise . 27 𝑥 3 + 125 [2] 3. (b) Explain why is the only real root of the operation . 𝑥 = − 5 3 27 𝑥 3 + 125 = 0 [2] 4. Express in partial fractions. 18 𝑥 + 7 𝑥 − 1 ( ) 4 𝑥 + 1 ( ) [4] 5. The polynomial is given by . 𝑓 ( 𝑥 ) 𝑓 ( 𝑥 ) = 2 𝑥 3 + 5 𝑥 2 − 4 𝑥 − 3 (a) Divide by . 𝑓 ( 𝑥 ) 𝑥 + 3 [2] (b) What can you deduce about ? 𝑥 + 3 [1] (c) Solve the equation . 𝑓 ( 𝑥 ) = 0 [2] 6. Factorise and explain why is the only real root of the equation 𝑥 3 − 8 𝑥 = 2 . 𝑥 3 − 8 = 0 [4]

7. By using long division, find the remainder when is 9 𝑥 4 − 13 𝑥 2 + 4 𝑥 − 2 divided by . 3 𝑥 2 + 2 𝑥 − 1 [3] 8. Given that for all real values of 20 𝑥 2 + 13 𝑥 + 5 = 𝐴𝑥 1 + 2 𝑥 ( ) + 𝐵𝑥 + 5 , find the value of and of . 𝑥 𝐴 𝐵 [3] 9. Express in partial fractions. 6 𝑥 𝑥 + 1 ( ) 𝑥 − 1 ( ) 2 [5] 10. Factorise . 125 𝑎 3 − 8 𝑏 3 [3] 11. The polynomial has a factor . 𝑓 ( 𝑥 ) = 6 𝑥 3 + 𝑎 𝑥 2 + 11 𝑥 − 6 𝑥 + 2( ) (a) Show that . 𝑎 = 19 [2] (b) Solve the equation Show your working clearly. 𝑓 ( 𝑥 ) = 0 . [4] (c) Hence solve the equation . 6 𝑦 3 + 19 𝑦 2 + 11 𝑦 − 6 = 0 [2] 12. Factorise . 8 𝑥 3 + 125 [3] 13. Express in partial fractions. 3 𝑥 2 − 2 𝑥 + 1 ( ) 2 2 𝑥 − 1 ( ) [5] 14. The polynomial is given by , where 𝑝 ( 𝑥 ) 𝑝 ( 𝑥 ) = 5 𝑥 3 + 𝑎 𝑥 2 + 𝑏𝑥 − 2 and are constants. It is given that is a factor of and when 𝑎 𝑏 𝑥 − 2( ) 𝑝 ( 𝑥 ) is divided by , the remainder is 5. 𝑝 ( 𝑥 ) 𝑥 − 1( ) (a) Show that and find the value of . 𝑎 = − 21 𝑏 [4] (b) Using the values from part (a), find the remainder when 𝑝 ( 𝑥 ) is divided by . 2 𝑥 + 1( ) [2] 15. (a) Divide by . 2 𝑥 3 − 𝑥 2 + 8 𝑥 − 4 2 𝑥 − 1 [1] 15. (b) Express in partial fractions. 2 + 5 𝑥 − 𝑥 2 2 𝑥 3 − 𝑥 2 + 8 𝑥 − 4 [5]

16. The polynomial is given by , where 𝑝 ( 𝑥 )