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प्रश्न
Use factor theorem to factorise the following polynominals completely.
x3 + 2x2 – 5x – 6
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उत्तर
Let f(x) = x3 + 2x2 – 5x – 6
Factors of (∵ 6 = ± 1 : ± 2, ± 3, ±)
Let x = –1, then
f(–1) = (–1)3 + 2(–1)2 – 5(–1) – 6
= –1 + 2(1) + 5 – 6
= –1 + 2 + 5 – 6
= 7 – 7
= 0
∵ f(–1) = 0
∴ x + 1 is a factor of f(x).
Similarly, (x - 2) and (x + 3) are the factors of f(x).
Since f(x) is a polynomial of degree 3.
So, it can not have more than three linear factors.
∴ f(x) = k(x + 1) (x - 2)(x + 3) ...(1)
⇒ x3 + 2x2 - 5x - 6
= k(x + 1)(x - 2)(x + 3)
Putting x = 0 on both sides, we get
-6 = k(1)(-2)(3)
⇒ k = 1
Putting k = 1 in equation (1), we get
f(x) = (x + 1)(x + 3)
Hence, x3 + 2x2 - 5x - 6 = (x + 1)(x - 2)(x + 3)
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