Using factor theorem, factorize each of the following polynomials:$x^4 - 7x^3 + 9x^2 + x- 10$

Given:

Given expression is $x^4 - 7x^3 + 9x^2 + x- 10$.

To do:

We have to find the given polynomial using factor theorem.

Solution:

Let $f(x)=x^{4}-7 x^{3}+9 x^{2}+7 x-10$

The factors of the constant term $-10$ are $\pm 1, \pm 2, \pm 5$ and $\pm 10$
Let $x=1$, this implies,

$f(1)=(1)^{4}-7(1)^{3}+9(1)^{2}+7(1)-10$

$=1-7+9+7-10$

$=17-17$

$=0$

Therefore $(x-1)$ is a factor of $f(x)$

Let $x=2$, this implies,

$f(2)=(2)^{4}-7(2)^{3}+9(2)^{2}+7(2)-10$

$=16-56+36+14-10$

$=66-66$

$=0$

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

Let $x=5$, this implies,

$f(5)=(5)^{4}-7(5)^{3}+9(5)^{2}+7(5)-10$

$=625-875+225+35-10$

$=885-885$

$=0$

Therefore $x-5$ is a factor of $f(x)$

Hence, $f(x)=(x-1)(x-2)(x-5)$.

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Updated on: 10-Oct-2022

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