<p> Algebra Worksheet 7 – Factor / remainder theorem</p><p>1. Using long division and synthetic division: a) Divide x 3 + 2x 2 + 3x + 4 by x + 2 b) Divide 2x 3 - x 2 + 3x - 2 by x + 3 c) Divide a 3 + 64b3 by a + 4b</p><p>2. Find the remainder when x 3 + 3x 2 - x + 7 is divided by 3x - 1</p><p>3. If 3x 3 - ax 2 + 2x + 1 has a remainder of –3 on division by x + 2 , find a.</p><p>4. Find both values of ‘a’ for which the polynomials 3x 2 - 7x - 4 and 2x 2 - 10x + 6 give the same remainder on division by x - a .</p><p>5. Factorise: a) x 2 - 7 d) x 3 + 4x 2 - 3x - 12 b) x 3 - 8x 2 + 11x + 20 e) 4x 3 - 7x 2 + 2x + 1 c) x 3 - 3x 2 - 10x + 24 f) 3x 3 + 7x 2 - 22x - 8</p><p>6. If x - 1 is a factor of f (x) = x 3 + bx 2 + bx - 15, find b.</p><p>7. If x + 2 is a factor of f (x) = 2x 4 + 3x 3 - cx - 33x - 18, find c.</p><p>8. Show 2x 2 + 3x + 1 is a factor of 2x 4 - 5x 3 - 5x 2 + 5x + 3.</p><p>9. A polynomial has a remainder of 2 when divided by x, and a remainder of –5 when divided by x - 4 . Find the remainder when the polynomial is divided by x 2 - 4x. (Delta)</p><p>10. Determine the value of the real number p so that the real roots of the equation 2x 2 - 12x + p + 2 = 0 are of the form a, a+2 (U.B.)</p><p>11. Find the remaining root for each of these cubics, given the first two roots. 3 2 1 a) 2x + x - 5x + 2 first two roots: 2 , -2 b) 2x 3 + 9x 2 + x - 12 first two roots: -4, 1 3 2 1 c) 6x + 17x - 26x + 8 first two roots: 2 , -4 3 2 3 d) - 6x +17x + 15x - 36 first two roots: - 2, 3 (Delta) 2x + 5 b 12. Given = a + , find a and b. x + 2 x + 2</p><p>13. Factorise g(x) = x 3 + 2x 2 - 19x - 20 and sketch the graph</p><p>Delta; p14-16: Exercises 2.2 – 2.4</p><p>Mathematics with Calculus Page 1 JM</p>