QQuestion
It is given that (x − 2) is a factor of 2x³ − 7x² + kx − 2.
(a) Find k.
(b) Hence, factorise the resulting polynomial completely.
✓Answer
(a) Since (x − 2) is a factor, substituting x = 2 gives remainder zero.
2(2)³ − 7(2)² + 2k − 2 = 0
⇒ 16 − 28 + 2k − 2 = 0
⇒ −14 + 2k = 0
⇒ 2k = 14
⇒ k = 7
(b) The polynomial becomes:
2x³ − 7x² + 7x − 2
Dividing by (x − 2) gives 2x² − 3x + 1.
Therefore:
2x³ − 7x² + 7x − 2 = (x − 2)(2x² − 3x + 1)
= (x − 2)(2x² − 2x − x + 1)
= (x − 2)[2x(x − 1) − 1(x − 1)]
= (x − 2)(2x − 1)(x − 1)
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