Use the factor theorem to show that (2x + 3) is a factor of 8x³ + 4x² - 10x + 3.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

1. M1 Substitute -3/2 and simplify
2. A1 Obtain -27+9+15+3 or equivalent, hence zero and conclude, may have explanation at start of working
3. Allow use of identity assuming a factor of 2x +3 to obtain a quadratic factor. Need to see use of 4 equations to verify quadratic for M1, A1 for conclusion. Allow verification by expansion.
4. Allow use of identity including a remainder to obtain a quadratic factor and a remainder of zero. Need to see use of 4 equations for M1, A1 for conclusion. Allow verification by expansion.
5. Allow use of long division, must reach a remainder of zero for M1
6. Need powers of -3/2 evaluating for A1
7. AG; necessary detail needed
8. B1 Use cos 2θ = 2cos² θ-1
9. M1 Simplify acos²θ + b = (6 cos θ – 5) / (2 cos θ + 1) to polynomial form
10. A1 Obtain 8 cos³ θ + 4cos² θ – 10cosθ + 3 = 0
11. AG; necessary detail needed, must be completely correct with no poor use of brackets for A1
12. *M1 Attempt either division by 2x+3 and reach partial quotient x² + kx or use of identity or inspection
13. A1 Obtain quotient 4x2 − 4x +1
14. A1 Obtain factorised form (2x+3)(2x−1)2
15. M1 Solve for cosθ=k to find at least one value between 0 and 360
16. A1 Obtain 60 and 300 and no others
17. Or equivalent using cosθ or c
18. Or equivalent
19. Or equivalent, may be implied by later work
20. Dependent *M
21. SC1: Equation solver used to obtain 60 and 300 and no others, then 5/5
22. SC2:Equation solver used to obtain 60 then 4/5
23. SC 3: cosθ=0.5, (cosθ=−1.5) seen implies first 3 marks.