It is given that f(x) = (x−a)²g(x), where f(x) and g(x) are polynomials. Show that (x-a) is a factor of f'(x) .

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

1. [1] (a) Use correct product rule. f'(x)=(x-a)²g'(x)+2(x-a)g(x). Allow incorrect chain rule.
2. [1] (a) Obtain correct derivative and state a clear conclusion. E.g. take out a factor of (x – a) and show the factorised form as far as (x - a)h(x) correctly (no further comment needed), or show that f'(a) = 0 and state that ‘(x – a) is a factor'.
3. [1] (b) Use f (3) = 0. (i.e., 2(3)³-4(3)² +3p + q = 0).
4. [1] (b) Obtain 3p+q=-18. OE. Powers should be evaluated but do not need to be simplified.
5. [1] (b) Use f'(3) = 0. (i.e., 6(3)²-8(3)+p = 0 or 54-24+p=0).
6. [1] (b) Obtain p = -30. OE.
7. [1] (b) Obtain p= -30, q = 72. Correct only.
8. [1] (b) Alternative Method: Attempt division of f(x) by (x-3)² as far as (2x+Q), where Q≠0.
9. [1] (b) Alternative Method: Obtain quotient (2x+8).
10. [1] (b) Alternative Method: Equate linear remainder to zero and compare coefficients, or expand (2x+8)(x-3)² and compare coefficients. Method to obtain an equation in p or q.
11. [1] (b) Alternative Method: Obtain one of p=-30, q=72.
12. [1] (b) Alternative Method: Obtain p=-30, q = 72. Correct answers only.
13. [1] (b) Alternative Method 3: f'(x)=6x²-8x+p.
14. [1] (b) Alternative Method 3: Use the factor x-3 in f'(x), i.e., f'(x)=(x-3)(6x+10).
15. [1] (b) Alternative Method 3: Obtain p = -30.
16. [1] (b) Alternative Method 3: Use the factor (x−3)² in f(x), i.e., f(x)=(x-3)²(2x+Q).
17. [1] (b) Alternative Method 3: Obtain Q=8, q=72.