Question
Using factor theorem, factorize the following polynomials:
$x^3 - 2x^2 - x + 2$
$x^3 - 2x^2 - x + 2$
✓
Answer
Let $f(x)=x^3-2 x^2-x+2$
The factors of constant term in $f(x)$ are $\pm 1, \pm 2$.
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,
$f(x)=k(x-1)(x+1)(x-2)$
$x^3-2 x^2-x+2=k(x-1)(x+1)(x-2) \ldots$
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,
$x^3-2 x^2-x+2=(x-1)(x+1)(x-2)$
The factors of constant term in $f(x)$ are $\pm 1, \pm 2$.
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,
$f(x)=k(x-1)(x+1)(x-2)$
$x^3-2 x^2-x+2=k(x-1)(x+1)(x-2) \ldots$
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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