Factorization Of Polynomials — Maths STD 9 — Question
Gujarat BoardEnglish MediumSTD 9MathsFactorization Of Polynomials4 Marks
Question
Using factor theorem, factorize the following polynomials:$ x^3 - 2x^2 - x + 2$
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Answer
Let $f(x) = x^3 - 2x^2 - x + 2$
The factors of constant term in f(x) are $\pm1,\pm2.$
We have $f(1)=1-2-1+2=0$
$\Rightarrow(x-1)$ is a factor of $f(x) f(-1)=-1-2+1+2=0$
$\Rightarrow(x+1)$ is a factor of $f(x) f(2)=8-8-2+2=0$
$\Rightarrow(x-2)$ is a factor of $f(x)$ Since $f(x)$ is a polynomial of degree $3.$
So, it cannot have more than 3 linear factors.
Thus, factors of $f(x)$ are $(x-1),(x+1)$ and $(x-2)$.
Therefore, $f(x)=k(x-1)(x+1)(x-2) x^3-2 x^2-x+2$
$=k(x-1)(x+1)(x-2) \ldots(1)$ Putting $x=0$ on both sides,
we get, $2=k(-1)(1)(-2) 2=2 k k=1$
Substituting $k =1$ in (1),
we get, $x^3-2 x^2-x+2=(x-1)(x+1)(x-2)$
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