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प्रश्न
Factorise:
2x3 – 3x2 – 17x + 30
उत्तर
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)
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