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Polynomials Higher Maths Polynomials introduction Polynomials 1 Factors Ans Polynomials Ans Curves cutting the x and the y axis Ans Quotient and remainder Factors of the form ax + b Ans Polynomial problems 1 Click on a topic Polynomial problems 2 Polynomial exam level questions Past Paper questions Ans 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) 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 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 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. 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) 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 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) 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) Division by ax + b Find the quotient and remainder in each of the following exercises. 1. 2. 3. 4. 5. 6. 7. 8. 9. 4x2 + 6x - 2 divided by 2x - 1 4x3 - 2x2 + 6x - 1 divided by 2x - 1 6x2 - 5x + 2 divided by 3x - 1 9x2 - 6x - 10 divided by 3x + 1 3x3 + 5x2 - 11x + 8 divided by 3x - 1 2x3 + 7x2 - 5x + 4 divided by 2x + 1 2x3 - x2 - 1 divided by 2x + 3 5x3 + 21x2 + 9x - 1 divided by 5x + 1 6x3 + x2 + 1 divided by 2x - 3 Solutions on next slide Division by ax + b Solutions 1. 2. 3. 4. 5. 6. 7. 8. Quotient Remainder 2x + 4 2 2x2 + 3 2 2x - 1 1 3x - 3 -7 x2 + 2x - 3 5 x2 + 3x - 4 8 x2 - 2x + 3 -10 x2 + 4x + 1 -2 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. 4. Show that x=2 is a root of the equation x3 + 5x2 - 4x – 20 = 0 and find the other roots. 5. 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. Polynomials problems 2 1. Find k if x+3 is a factor of x3 – 3x2 + kx + 6 2. Find p if x4 + 4x3 + px2 + 4x + 1 has x+1 as a factor. Hence factorize fully. 3. If x+3 and x-1 are factors of f(x) = x4 + 2x3 - 7x2 + ax + b , find a and b and hence factorize fully. 4. If x+2 is a factor of x3 + kx2 - x – 2 , find k and hence factorize fully. 5. If x=3 is a root of the equation x3 – 37x + k = 0, find k and hence find all the other roots. 6. 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. 7. 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. Revision - Exam level questions 1. 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. 4. Find the points where the curve y = 4x3 – 4x2 - 29x + 15 cuts the x-axis. 5. Factorize fully a) 2x3- 3x2 - 11x + 6 b) 3x3 - 2x2 - 19x - 6 6. x3 + kx2- 13x - 10 is divisible by x + 2. Find the value of k. 7. 2x3 - 9x2 + ax + 30 is divisible by 2x - 3. Find a. 8. x + 3 is a factor of 3x3+ 2x2 + nx + 6. 9. Find n then factorize fully. x4 - 2x3 + kx2 + 3x - 2 has x + 2 as a factor. Find the value of k. 10. 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. Polynomials - (Questions from past papers) 1. Factorise fully 2x3 – 3x2 - 11x + 6 2. Factorise fully x3 – 6x2 + 9x – 4 3. Factorise fully 2x3 + 5x2 - 4x – 3 4. Find ‘p’ if (x+3) is a factor of x3 – x2 + px + 15. 5. 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 6. One root of 2x3 – 3x2 + px + 30 = 0 is x = - 3. Find ‘p’ and hence find the other roots. 7. 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 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. 10. 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. 12. 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 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