Question

Factorise:

2x³ – 3x² – 17x + 30

Answer 1

Let p(x) = 2x³ – 3x² – 17x + 30

Constant term of p(x) = 30

∴ Factors of 30 are ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30

By trial, we find that p(2) = 0, so (x – 2) is a factor of p(x)  ...[∵ 2(2)³ – 3(2)² – 17(2) + 30 = 16 – 12 – 34 + 30 = 0]

Now, we see that 2x³ – 3x² – 17x + 30

= 2x³ – 4x² + x² – 2x – 15x + 30

= 2x²(x – 2) + x(x – 2) – 15(x – 2)

= (x – 2)(2x² + x – 15)  ...[Taking (x – 2) common factor]

Now, (2x² + x – 15) can be factorised either by splitting the middle term or by using the factor theorem.

Now, (2x² – x – 15) = 2x² + 6x – 5x – 15  ...[By splitting the middle term]

= 2x(x + 3) – 5(x + 3)

= (x + 3)(2x – 5)

∴ 2x³ – 3x² – 17x + 30 = (x – 2)(x + 3)(2x – 5)