Using factor theorem, factorize each of the following polynomials:$x^3 - 6x^2 + 3x + 10$

Given:

Given expression is $x^3 - 6x^2 + 3x + 10$.

To do:

We have to find the given polynomial using factor theorem.

Solution:

The factors of the constant term 10 are $\pm 1, \pm 2, \pm 5, \pm 10$

Let $x=-1$, this implies,

$f(-1)=(-1)^{3}-6(-1)^{2}+3(-1)+10$

$=-1-6-3+10$

$=10-10$

$=0$

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

Let $x=-2$, this implies,

$f(-2)=(-2)^{3}-6(-2)^{2}+3(-2)+10$

$=-8-24-6+10$

$=-38+10$

$=-28$

Therefore $x+2$ is not a factor of $f(x)$.

Let $x=2$, this implies,

$f(2)=(2)^{3}-6(2)^{2}+3 \times 2+10$

$=8-24+6+10$

$=24-24$

$=0$

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

Let $x=5$, this implies,

$f(5)=(5)^{3}-6(5)^{2}+3 \times 5+10$

$=125-150+15+10$

$=150-150$

$=0$

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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