O-LevelAdditional MathematicsFactors of polynomials4037/12 · May/June 2025 · Q510 Marks
The polynomial p is such that p(x) = 3x³ − 7x² + ax + b, where a and b are integers. It is given that p'(-1) = 21 and that x-2 is a factor of p(x).
- (a)Find the values of a and b.[4]
- (b)Hence write p(x) as a product of linear factors with integer coefficients.[3]
- (c)Using your values of a and b, solve the equation 3e⁶ʸ − 7e⁴ʸ + ae²ʸ + b = 0.[3]
Mark scheme
Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.
- [B1] p'(x)=9x²-14x+a_soi
- [B1] p'(-1) = 9+14+ a = 21
- [B1] a=-2
- [B1] p(2) = 24-28 + 2a+b=0_soi (Each term must be simplified, allow using their a)
- [B1] b=8
- [M1] [x-2](3x²−x−4) soi (for quadratic factor with two terms correct)
- [A1] (x-2)(x+1)(3x-4) (must be from correct a and b)
- [B1] (e²ʸ-2) = 0 soi
- [B2] y = 1/2 ln(4/3) oe, y = 1/2 ln 2 oe (for both solutions and no extra solution)
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