O-LevelAdditional MathematicsFactors of polynomials4037/12 · Oct/Nov 2016 · Q99 Marks
Do not use a calculator in this question. The polynomial p(x) is ax³ - 4x² + bx + 18. It is given that p(x) and p'(x) are both divisible by 2x-3.
- (i)Show that a = 4 and find the value of b.[4]
- (ii)Using the values of a and b from part (i), factorise p(x) completely.[2]
- (iii)Hence find the values of x for which p(x) = x + 2 .[3]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2016). The mark or mark type is shown in brackets.
- [1] (i) p(3/2) = a(3/2)³ - 4(3/2)² + b(3/2) + 18 (=0) M1 for attempt at p(3/2)
- [1] (i) p'(x) = 3ax² - 8x + b; p'(3/2) = 3a(3/2)² - 8(3/2) + b (=0) M1 for differentiation and attempt at p'(3/2)
- [1] (i) Simultaneous equations: 9a + 4b + 24 = 0 and 27a + 4b - 48 = 0 M1 for solution of simultaneous equations, to get either a or b
- [1] (i) a = 4, b = -15 A1 for both
- [2] (ii) (x+2)(2x-3)² oe M1 for attempt at long division or factorisation, A1 for correct factorisation
- [1] (iii) From p(x) = x+2, one solution is x = -2 B1 Must be using (x + 2) correctly using part (ii) to get x = -2
- [1] (iii) (2x-3)² = 1 M1 for solution of the quadratic equation
- [1] (iii) x = 1, x = 2 A1
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