The polynomial p(x) is defined by p(x) = 2x⁴ + kx³ + kx² + 17x + 18, where k is a constant. It is given that (x+2) is a factor of p(x).

Mark scheme

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

1. [M1] (a) Substitute x=−2, equate to zero and attempt solution
2. [A1] (a) Obtain 32-8k+4k-34+18=0 or equivalent and hence k=4
3. [B1] (b) State α=−2
4. [M1] (b) Divide by x+2 at least as far as 2x³ + mx [Or equivalent method, e.g. inspection.]
5. [A1] (b) Obtain 2x³ + 4x +9
6. [A1] (b) Rearrange 2x³ +4x+9=0 to confirm x = √(-2x-4.5) [AG necessary detail needed.]
7. [M1] (c) Consider sign of x - √(-2x-4.5) or equivalent for –1.4 and −1.0 [Or Example 1: f(x)=2x³ +4x+9, finding f(−1.4)and f(−1). Example 2: f(x)=x - √(-2x-4.5), finding f(−1.4) and f(−1).]
8. [A1] (c) Obtain -0.2... and 0.3... or equivalents and justify conclusion [Example 1: f(−1)=3, f(−1.4)=-1.2... with a correct conclusion. Example 2: f(−1)=−1.357.... so −1>−1.357... f(−1.4)= −1.193.... so −1.4<−1.19... with a correct conclusion. AG - necessary detail needed.]
9. [M1] (d) Use iterative process correctly at least once
10. [A1] (d) Obtain final answer -1.26 [Answer required to exactly 3 significant figures.]
11. [A1] (d) Show sufficient iterations to 5 sf to justify answer or show sign change in interval [-1.265, -1.255]