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1. Polynomials Higher Maths

2. Polynomials introduction Polynomials 1 Factors Curves cutting the x and the y axis Quotient and remainder Polynomials Ans Ans Ans Ans Ans Polynomial problems 1 Click on a topic Past Paper questions Polynomial exam level questions Factors of the form ax + b Polynomial problems 2

3. Polynomials (Introduction) Use nested multiplication to find the values of the functions below: 1.   f(x) = x3 + 2x2 + 3x + 5 , find f(3) 2.   f(x) = 3x2 - 4x + 7 , find f(2) 3.   f(x) = 2x3 - x2 – x - 1 , find f(-2) 4.   f(x) = x4 + 2x3 – x2 + x + 1 , find f(3) 5.   f(x) = x3 - 2x2 - 3x - 7 , find f(1) 6.   f(x) = x4 - x3 + x2 + x + 2 , find f(4) 7.   f(x) = x4 - 2x2 - 2 , find f(-1) 8.   f(x) = 2x5 + x3 - 6x + 8 , find f(2) 9.   f(x) = 7x2 - 2x + 3 , find f(-1) 10.   f(x) = x3 - 10x2 - 8 , find f(-2) 11.   f(x) = 2x5 + 3x4 - x2 + 2x - 3 , find f(-1) 12.   f(x) = x6 + 2x4 -3x3 + 2x2 + 1 , find f(-3)

4. Polynomials (1) Use nested multiplication to find the values of the functions below 1  f(x) = 2x3 - 3x2 + 5x + 1 , find f(4) 2    f(x) = x4 - x3 + 2x2 + x + 3 , find f(-2) 3  f(x) = 3x3 - 4x2 + 5 , find f(2) 4  f(x) = 2x4 + 3x3 -8x + 5 , find f(3) 5 f(x) = 2x5 + 3x3 + 4x2 - 7x - 1 , find f(1) 6  f(x) = x4 - 6x3 + 3x2 + 4x + 2 , find f(-2) 7  f(x) = 4x3 - 7x2 - x - 2 , find f(-1) 8 f(x) = 2x4 + 4x3 - 6x + 8 , find f(1/2) 9  f(x) = 8x2 - 2x + 3 , find f(-1/2) 10    f(x) = x3 - 10x2 + 5x - 8 , find f(-2) 11 f(x) = 2x5 + 3x4 - x2 - 2x + 3 , find f(-1) 12  f(x) = x6 + 2x4 -3x3 + 2x2 + 1 , find f(-3) Solutions on next slide

5. Solutions 1  f(4) = 101  2 f(-2) = 33 3  f(2) = 13 4 f(3) = 224 5 f(1) = 1 6 f(-2) = 70 7 f(-1) = -12  8 f(1/2) = 5.625 9 f(-1/2) = 6 10 f(-2) = -66 11 f(-1) = 5  12 f(-3) = 991

6. Factors 1 Show that x - 2 is a factor of x3 + x2 - 10x + 8 and hence factorise fully. 2 Show that x - 4 is a factor of x3 - 4x2 - 9x + 36 and hence factorise fully. 3 Show that x + 2 is a factor of x3 + 4x2 + x - 6 and hence factorise fully. 4 Show that x + 1 is a factor of x3 - 6x2 + 3x + 10 and hence factorise fully. 5 Show that x - 2 is a factor of 2x3 - 7x2 + 7x - 2 and hence factorise fully. 6 Show that x + 4 is a factor of 3x3 + 14x2 + 7x - 4 and hence factorise fully. 7 Show that x + 3 is a factor of x3 + 3x2 - 25x - 75 and hence factorise fully. 8 Show that x - 3 is a factor of 4x3 - 21x2 + 29x - 6 and hence factorise fully. 9 Show that x - 1 is a factor of 8x3 - 14x2 + 7x - 1 and hence factorise fully.

7. Factors  - Some Solutions 1 (x - 2)(x - 1)(x + 4) 2 (x + 3)(x - 3)(x - 4) 3 (x - 1)(x + 3)(x + 2) 4 (x + 1)(x - 2)(x - 5) 5  (x - 2)(2x - 1)(x - 1)

8. Curves cutting the x and y axes In each example, find the points where the curve cuts the x and y axes. 1.   y = x3 + x2 - 10x + 8 2.   y = x3 + 6x2 + 11x + 6    3.   y = x3 - 8x2 + 17x - 10    4.   y = x3 - x2 - 10x - 8    5.   y = x3 + 4x2 + x - 6    6.   y = x3 + x2 - 16x + 20    7.   y = 2x3 - 17x2 + 22x - 7    8.   y = 6x3 - 17x2 + 11x - 2    9.   y = 4x3 + 11x2 + 5x - 2    10.   y = 3x3 - 5x2 - 4x + 4    11.   y = x3 - 8x2 + 11x + 20    12.   y = 2x3 - x2 - 15x + 18    13.   y = x4 - 5x3 + 5x2 + 5x - 6    14.   y = 2x4 + 9x3 + 6x2 - 11x - 6 Solutions on next slide

9. Curves cutting the x and y axes - Solutions 1.   (0, 8)   (-4, 0)    (2, 0)    (1, 0)    2.   (0, 6)   (-1, 0)    (-2, 0)    (-3, 0)    3.   (0, -10)   (1, 0)    (2, 0)    (5, 0)    4.   (0, -8)   (-2, 0)    (-1, 0)    (4, 0)    5.   (0, -6)   (1, 0)    (-2, 0)    (-3, 0)    6.   (0, 20)   (2, 0)    (2, 0)    (-5, 0) 7.   (0, -7)   (7, 0)    (½, 0)    (1, 0)  8.   (0, -2)   (2, 0)    (½, 0)    (1/3, 0)    9.   (0, -2)   (¼, 0)    (-1, 0)    (-2, 0)    10.   (0, 4)   (2/3, 0)    (-1, 0)    (2, 0)    11.   (0, 20)   (4, 0)    (5, 0)    (-1, 0)    12.   (0, 18)   (3/2, 0)    (2, 0)    (-3, 0)    13.   (0, -6)   (1,0)    (2, 0)    (3, 0)    (-1, 0)    14.   (0, -6)   (-½, 0)   (1, 0)    (-2, 0)    (-3, 0)

10. Quotient and Remainder Find the quotient and the remainder in each example. 1 x2 – 5x + 2 ÷ (x – 3) 2 2x2 + x + 3 ÷ (x – 1) 3 x3 + x2 – 3x + 1 ÷ (x + 2) 4 x3 - 2x2 – x - 3 ÷ (x + 1) 5 x2 – x - 2 ÷ (x – 1) 6 2x2 - 3x - 4 ÷ (x + 3) 7 x3 - 4x2 – 4x + 16 ÷ (x - 4) 8 x2 – 6x - 7 ÷ (x – 7) 9 x2 + x + 5 ÷ (x – 2) 10 x3 + 2x2 – 4 ÷ (x - 1) 11 2x3 - 3x2 – 4x + 7 ÷ (x + 1) 12 x3 - 4x2 – 7x + 10 ÷ (x - 5) 13 x4 + x2 + 1 ÷ (x - 2) 14 x3 - x2 + x - 1 ÷ (x - 1)

11. Division by ax + b Find the quotient and remainder in each of the following exercises.  1.   4x2 + 6x - 2 divided by 2x - 1    2.   4x3 - 2x2 + 6x - 1 divided by 2x - 1    3.   6x2 - 5x + 2 divided by 3x - 1    4.   9x2 - 6x - 10 divided by 3x + 1    5.   3x3 + 5x2 - 11x + 8 divided by 3x - 1    6.   2x3 + 7x2 - 5x + 4 divided by 2x + 1    7.   2x3 - x2 - 1 divided by 2x + 3    8.   5x3 + 21x2 + 9x - 1 divided by 5x + 1    9.   6x3 + x2 + 1 divided by 2x - 3 Solutions on next slide

12. Division by ax + b Solutions      Quotient      Remainder 1.   2x + 4      2     2.    2x2 + 3      2    3.   2x - 1      1    4.  3x - 3      -7    5.    x2 + 2x - 3       5    6.   x2 + 3x - 4      8    7.    x2 - 2x + 3     -10    8.   x2 + 4x + 1       -2

13. Polynomials Problems 1 • 1. Show that x-4 is a factor of 2x2 – 11x + 12 and hence factorize fully. • 2. Factorize fully x3 – 11x2 + 26x – 16 • 3. If x+3 is a factor of x3 + kx2 + 7x + 3 , find k and hence factorize fully. • Show that x=2 is a root of the equation x3 + 5x2 - 4x – 20 = 0 and • find the other roots. • Find the points where the curve y = x3 + 10x2 - 9x – 90 cuts • the coordinate axis. • 6. Factorize fully x3 + 2x2 - x – 2. • 7. If x-1 is a factor of x3 - 3x2 + kx – 1, find k and hence factorize fully. • 8. Show that x=1 is a root of the equation x3 - 9x2 + 20x–12 = 0 • and find the other roots. • 9. Show that x =-4 is a root of the equation 6x3 + 25x2 + 2x–8 = 0 • and find the other roots. • 10. If x-2 is a factor of f(x) = 2x3 + kx2 + 7x + 6 , find k and hence • solve the equation f(x) = 0 with this value of k. • 11. The same remainder is obtained when x2 + 3x – 2 and • x3 - 4x2 + 5x + p are divided by x+1. Find p.

14. Polynomials problems 2 • Find k if x+3 is a factor of x3 – 3x2 + kx + 6 • Find p if x4 + 4x3 + px2 + 4x + 1 has x+1 as a factor. • Hence factorize fully. • If x+3 and x-1 are factors of f(x) = x4 + 2x3 - 7x2 + ax + b , find a and b • and hence factorize fully. • If x+2 is a factor of x3 + kx2 - x – 2 , find k • and hence factorize fully. • If x=3 is a root of the equation x3 – 37x + k = 0, find k • and hence find all the other roots. • Given that x-2 is a factor of f(x) = 2x3 + kx2 + 7x + 6, find k. • Hence solve the equation f(x) = 0 with this value of k. • Find k if 2x3 + x2 + kx – 8 is divisible by x+2. • 8. Find k if x3 + kx2 - 6x + 8 has a factor x-4. • Hence factorize the expression fully.

15. Revision ‑ Exam level questions • 2x + 1 is a factor of 2x3 – tx2 + x + 2. Find t. • 2. If x + 1 and x ‑ 3 are factors of f(x) = 2x3 ‑ 5x2 + px + q, find p and q. • 3. Given that 2x ‑ 1 is a factor of 4x3‑ 4x2 + kx + 15 , find k. • Factorize fully when k has this value. • Find the points where the curve y = 4x3 – 4x2 - 29x + 15 cuts the x‑axis. • Factorize fully a) 2x3‑ 3x2 ‑ 11x + 6 b) 3x3 ‑ 2x2‑ 19x ‑ 6 • x3 + kx2‑ 13x ‑ 10 is divisible by x + 2. Find the value of k. • 2x3 ‑9x2 + ax + 30 is divisible by 2x ‑ 3. Find a. • x + 3 is a factor of 3x3+ 2x2 + nx + 6. Find n then factorize fully. • 9. x4 - 2x3 + kx2 + 3x ‑ 2 has x + 2 as a factor. Find the value of k.

16. Factorize fully x3+ 6x2+ 9x + 4 and hence solve • x3+ 6x2+ 9x + 4 = 0.Find the stationary points on the • curve y = x3+ 6x2+ 9x + 4 and determine their nature. Sketch the curve. • 11. If x ‑ 1 and x + 3 are both factors of 2x3+ ax2 + bx + 3, find the values of a and b. • 12. Find k if x + 1 is a factor of x3 + kx2 ‑ 5x ‑ 6. Find the other factors when k has this value. • 13. Solve the equation x3‑ x2 + x ‑ 6 = 0. Hence find the equation of the tangent to the curve y = x3‑ x2 + x ‑ 6 at the points where it cuts the x‑axis. Find the equation of the tangent at the point where the curve crosses the y‑axis. Show that the two tangents meet at (3 /2, ‑9/2). • 14. If f(x) = 3x4 + 8x3 – 6x2 , solve the equation f '(x) = 24.

17. Polynomials - (Questions from past papers) • Factorise fully 2x3 – 3x2 - 11x + 6 • Factorise fully x3 – 6x2 + 9x – 4 • Factorise fully 2x3 + 5x2 - 4x – 3 • Find ‘p’ if (x+3) is a factor of x3 – x2 + px + 15. • When f(x) = 2x4 – x3 + px2 + qx + 12 is divided by (x – 2) • the remainder is 114. One factor of f(x) is (x + 1). Find p and q • One root of 2x3 – 3x2 + px + 30 = 0 is x = - 3. • Find ‘p’ and hence find the other roots. • Show that (x – 3) is a factor of f(x) = 2x3 + 3x2 – 23x – 12 and • hence factorise f(x) fully. Continued on the next slide

18. 8. Find a real root of the equation 2x3 – 3x2 + 2x – 8 = 0 • Show algebraically that there are no other real roots. • 9. Find ‘k’ if (x – 2) is a factor of x3 + kx2 – 4x – 12 • and hence factorise fully. • Express x3 - 4x2 – 7x + 10 in fully factorised form. • 11. Show that x = 2 is a root of the equation • 2x3 + x2 – 13x + 6 = 0 and hence find the other roots. • Given that (x + 2) is a factor of 2x3 + x2 + kx + 2, find the value of k. • Hence solve the equation 2x3 + x2 + kx + 2 = 0 when k takes this value. • 13. Given that (x – 2) and (x + 3) are factors • of f(x) = 3x3 + 2x2 + cx + d, find the values of ‘c’ and ‘d’. Solutions on next slide

19. Answers to – Polynomials past paper questions Question Solution 1 (x – 3)(2x – 1)(x + 2) 2 (x – 1)(x – 1)(x - 4) 3 (x – 1)(2x + 1)(x + 3) 4 p = -7 5 4p + 2q = 78; p – q = -15 solve to give p=8 and q = 23 6 Roots are x=-3 , 2 and 5/2 7 (x – 3)(2x + 1)(x + 4) 8 (x-2)(2x2+x+4) = 0 So x = 2 is a root and there are no more roots because you cannot take the square root of a negative number which occurs when you apply the quadratic formula. 9 K = 3 ; So this gives f(x) = (x – 2)(x + 3)(x + 2) 10 (x – 1)(x –5)(x + 2) 11 Roots are x = 2 , x = ½ and x = -3 12 k = -5 ; x = -2 , ½ and 1 13 2c + d + 32 = 0, -3c + d -63 = 0 solve to give c = -19 and d = 6