Using factor theorem, factorize each of the following polynomials:$x^4 - 2x^3 - 7x^2 + 8x + 12$

Given:

Given expression is $x^4 - 2x^3 - 7x^2 + 8x + 12$.

To do:

We have to factorize the given polynomial.

Solution:

Let $f(x)=x^{4}-2 x^{3}-7 x^{2}+8 x+12$

The factors of the constant term 12 are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.

Let $x=1$, this implies,

$f(1)=(1)^{4}-2(1)^{3}-7(1)^{2}+8(1)+12$

$=1-2-7+8+12$

$=21-9$

$=12 \
eq 0$

Therefore, $x-1$ is not a factor of $f(x)$.

Let $x=-1$, this implies,

$f(-1)=(-1)^{4}-2(-1)^{3}-7(-1)^{2}+8(-1)+12$

$=1+2-7-8+12$

$=15-15$

$=0$

Therefore, $x+1$ is a factor of $f(x)$.

Let $x=-2$, this implies,

$f(-2)=(-2)^{4}-2(-2)^{3}-7(-2)^{2}+8(-2)+12$

$=16+16-28-16+12$

$=44-44$

$=0$

Therefore, $x+2$ is a factor of $f(x)$.

Let $x=2$, this implies,

$f(2)=(2)^{4}-2(2)^{3}-7(2)^{2}+8(2)+12$

$=16-16-28-16+12$

$=44-44$

$=0$

Therefore, $x-2$ is a factor of $f(x)$.

Let $x=3$, this implies,

$f(3)=(3)^{4}-2(3)^{3}-7(3)^{2}+8(3)+12$

$=81-54-63+24+12$

$=117-117$

$=0$

Therefore, $x-3$ is a factor of $f(x)$

Hence, $f(x)=(x+1)(x+2)(x-2)(x-3)$.

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Updated on: 10-Oct-2022

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