O-LevelAdditional MathematicsFactors of polynomials4037/12 · Oct/Nov 2012 · Q109 Marks
A function f is such that f(x) = 4x³ + 4x² + ax + b. It is given that 2x – 1 is a factor of both f(x) and f'(x).
- (i)Show that b = 2 and find the value of a.[5]
- (ii)find the remainder when f(x) is divided by x + 3,[2]
- (iii)express f(x) in the form f(x) = (2x – 1)(px² + qx + r), where p, q and r are integers to be found,[2]
- (iv)find the values of x for which f(x) = 0.[2]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2012). The mark or mark type is shown in brackets.
- [M1] M1 for use of x = 1/2 and equating to zero
- [M1] M1 for differentiation
- [M1] M1 for attempt to obtain a = -7 from f'(1/2) = 0
- [A1]
- [A1]
- [M1] M1 for use of x = -3 in either the remainder theorem or algebraic long division.
- [A1]
- [M1, A1] M1 for attempt to obtain quadratic factor
- [B1] B1 for each – must be correct from work
- [B1]
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