Chesley

2020-12-24

Need to calculate:The factorization of ${x}^{3}+3{x}^{2}+2x+6$

dessinemoie

Formula used:
The factors of a polynomial can be find by taking a common factor and this method is called factor by grouping,
$ab+ac+bd+cd=a\left(b+c\right)+d\left(b+c\right)$
$=\left(a+d\right)\left(b+c\right)$
Or,
$ab—ac+bd—cd=a\left(b—c\right)+d\left(b-c\right)$
$=\left(a+d\right)\left(b-c\right)$
Calculation:
Consider the polynomial ${x}^{3}+3{x}^{2}+2x+6$.
This is a four term polynomial, factorization of this polynomial can be find by factor by grouping as,
${x}^{3}+3{x}^{2}+2x+6=\left(x3+3{x}^{2}\right)+\left(2x+6\right)$
$={x}^{2}\left(x+3\right)+2\left(x+3\right)$
As, $\left(x+3\right)$ is the common factor of the polynomial,
The polynomial can be factorized as,
${x}^{3}+3{x}^{2}+2x+6={x}^{2}\left(x+3\right)+2\left(x+3\right)$
$=\left(x+3\right)\left({x}^{2}+2\right)$
Therefore, the factorization of the polynomial ${x}^{3}+3{x}^{2}+2x+6$ is $\left(x+3\right)\left({x}^{2}+2\right)$.

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