Using factor theorem, factorize each of the following polynomials:$x^3 + 6x^2 + 11x + 6$

Given:

Given expression is $x^3 + 6x^2 + 11x + 6$.

To do:

We have to find the given polynomial using factor theorem.

Solution:

$f(x)=x^{3}+6 x^{2}+11 x+6$

The factors of constant term 6 are $\pm 1, \pm 2, \pm 3$ and $\pm 6$

Let $x=-1$, this implies,

$f(-1)=(-1)^{3}+6(-1)^{2}+11(-1)+6$

$=-1+6-11+6$

$=12-12$

$=0$

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

Let $x=-2$, this implies,

$f(-2)=(-2)^{3}+6(-2)^{2}+11(-2)+6$

$=-8+24-22+6$

$=30-30$

$=0$

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

Let $x=-3$, this implies,

$f(-3)=(-3)^{3}+6(-3)^{2}+11(-3)+6$

$=-27+54-33+6$

$=60-60$

$=0$

Therefore $x+3$ is a factor of $f(x)$

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

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Updated on: 10-Oct-2022

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