QQuestion
The polynomial kx³ + 3x² − 11x − 6 leaves a remainder of 6 when divided by x + 1.
(a) Find the value of k.
(b) Hence, factorise the polynomial completely.
ICSEClass XMathematicsFactorisation04 Mark2026 Hard
✓Answer
Let P(x) = kx³ + 3x² − 11x − 6.
Since P(x) leaves a remainder of 6 when divided by x + 1, by the Remainder Theorem:
P(−1) = 6
⇒ k(−1)³ + 3(−1)² − 11(−1) − 6 = 6
⇒ −k + 3 + 11 − 6 = 6
⇒ −k + 8 = 6
⇒ −k = −2
⇒ k = 2
Therefore, P(x) = 2x³ + 3x² − 11x − 6.
Now, P(2) = 2(2³) + 3(2²) − 11(2) − 6
⇒ P(2) = 16 + 12 − 22 − 6 = 0
Therefore, x − 2 is a factor.
Dividing 2x³ + 3x² − 11x − 6 by x − 2 gives:
Now, 2x² + 7x + 3
= 2x² + 6x + x + 3
= 2x(x + 3) + 1(x + 3)
= (2x + 1)(x + 3)
Hence, the complete factorisation is:
2x³ + 3x² − 11x − 6 = (x − 2)(2x + 1)(x + 3).
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