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प्रश्न
If (x – 2) is a factor of 2x3 – x2 + px – 2, then
(i) find the value of p.
(ii) with this value of p, factorise the above expression completely
उत्तर
(i) Let x – 2 = 0,
then x = 2
Now f(x) = 2x3 – x2 + px – 2
∴ f(2) = 2(2)3 – (2)2 + p x 2 – 2
= 2 x 8 – 4 + 2p – 2
= 16 – 4 + 2p – 2
= 10 + 2p
(ii) ∴ f(2) = 0,
then 10 + 2p = 0
⇒ 2p = –10
⇒ p = –5
Now, the polynomial will be
2x3 – x2 – 5x – 2
= (x – 2)(2x2 + 3x + 1)
= ( x – 2)[2x2 + 2x + x + 1]
= (x – 2)[2x(x + 1) + 1(x + 1)]
= (x – 2)(x + 1)(2x + 1)
`x – 2")"overline(2x^3 – x^2 – 5x – 2)("2x^2 + 3x + 1`
2x3 – 4x2
– +
3x2 – 5x
3x2 – 6x
– +
x – 2
x – 2
– +
x
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