Using factor theorem, factorize of the polynomials:
$x^{3}+2 x^{2}-x-2$
Given, $f(x)=x^{3}+2 x^{2}-x-2$
The constant term in f(x) is -2
The factors of (-2) are ±1, ± 2
Let, x – 1 = 0
=> x = 1
Substitute the value of x in f(x)
$f(1)=(1)^{3}+2(1)^{2}-1-2$
= 1 + 2 – 1 – 2
= 0
Similarly, the other factors (x + 1) and (x + 2) of f(x)
Since, f(x) is a polynomial having a degree 3, it cannot have more than three linear factors.
∴ f(x) = k(x – 1)(x + 2)(x + 1 )
$x^{3}+2 x^{2}-x-2=k(x-1)(x+2)(x+1)$
Substitute x = 0 on both the sides
0 + 0 – 0 – 2 = k(-1)(1)(2)
=> – 2 = - 2k
=> k = 1
Substitute k value in f(x) = k(x – 1)(x + 2)(x + 1)
f(x) = (1)(x – 1)(x + 2)(x + 1)
=> f(x) = (x – 1)(x + 2)(x + 1)
So, $x^{3}+2 x^{2}-x-2=(x-1)(x+2)(x+1)$
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