O-LevelAdditional MathematicsFactors of polynomials4037/12 · Oct/Nov 2025 · Q27 Marks
The polynomial p is such that p(x) = 2x³ + ax² + 13x+b, where a and b are integers. It is given that x+2 is a factor of p(x). When p(x) is divided by x+1 there is a remainder of 6.
- (a)Find the values of a and b.[4]
- (b)Show that the equation p(x) = 0 has only one real root.[3]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.
- [4] B1 (p(-2)=) -16+4a-26+b=0
- B1 (p(-1)=) -2+a-13+b=6
- M1 to solve their equations to obtain at least one value, must be using p(-2) or p(-1)
- A1 a=7, b=14
- [3] M1 for attempt to factorise or use of long division on their p(x)
- B1 FT on their quadratic factor. Quadratic factor must have a negative discriminant and have a correct value if evaluated
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