A-LevelMathematicsAlgebra9709/31 · May/June 2025 · Q55 Marks
The polynomial 3x³ +pax²+7a²x+qa³ is denoted by f(x), where p, q and a are constants and a ≠ 0. When f(x) is divided by (x+2a) the remainder is -22a³. When f(x) is divided by (3x-a) the remainder is -a³. Find the values of p and q.
- The polynomial 3x³ +pax²+7a²x+qa³ is denoted by f(x), where p, q and a are constants and a ≠ 0. When f(x) is divided by (x+2a) the remainder is -22a³. When f(x) is divided by (3x-a) the remainder is -a³. Find the values of p and q.[5]
Mark scheme
Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.
- [M1] Use f(−2a) = -22a³ Or use long division and equate a constant remainder to -22a³.
- [A1] Obtain -24a³ + 4pa³ −14a³ + qa³ = −22a³ Must evaluate the terms. OE, e.g. 4p + q=16.
- [M1] Use f(a/3) = -a³ Or use long division and equate a constant remainder to -a³.
- [A1] Obtain (1/9)a³ + (1/9)pa³ + (7/3)a³ + qa³ = -a³ Must evaluate the terms. OE, e.g. p +9q =-31
- [A1] Obtain p = 5, q = −4
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