A-LevelMathematicsAlgebra9709/22 · Oct/Nov 2017 · Q47 Marks
The polynomials p(x) and q(x) are defined by p(x) = x³ + x² + ax - 15 and q(x) = 2x³ + x² + bx + 21, where a and b are constants. It is given that (x + 3) is a factor of p(x) and also of q(x).
- (i)Find the values of a and b.[3]
- (ii)Show that the equation q(x) - p(x) = 0 has only one real root.[4]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.
- [M1] Substitute x = -3 into either p(x) or q(x) and equate to zero (may be implied)
- [A1] Obtain a = -11
- [A1] Obtain b = -8
- [*M1] Divide x +3 into expression for q(x) - p(x) ( may be a four term cubic equation), or Obtain a 3 term cubic equation by subtraction
- [A1] Obtain x² - 3x+12 or x²-2x-5 and 2x²-5x+7
- [DM1] Apply discriminant to quadratic factor of q(x)-p(x) or equivalent
- [A1] Obtain -39 or equivalent and conclude appropriately
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