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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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