O-LevelAdditional MathematicsFactors of polynomials4037/11 · May/June 2016 · Q108 Marks
Given that f(x) = 4x³ + kx + p is exactly divisible by x + 2 and f'(x) is exactly divisible by 2x-1, find the value of k and of p.
- (i)Given that f(x) = 4x³ + kx + p is exactly divisible by x + 2 and f'(x) is exactly divisible by 2x-1, find the value of k and of p.[4]
- (ii)Using the values of k and p found in part (i), show that f(x) = (x + 2)(ax² + bx + c), where a, b and c are integers to be found.[2]
- (iii)Hence show that f(x) = 0 has only one solution and state this solution.[2]
Mark scheme
Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.
- [1] f(-2) = -32 - 2k + p = 0
- [1] f'(x) = 12x² + k
- [1] f'(1/2) = 12(1/2)² + k = 0 => 3 + k = 0 => k = -3
- [1] leading to k = -3 and p = 26
- [1] (x+2)(4x² - 8x + 13)
- [1] 4x² - 8x + 13
- [1] Showing that 4x² - 8x + 13 = 0 has no real roots
- [1] so x = -2 only www
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