defazajx

2021-03-09

Need to calculate:The factorization of ${x}^{3}+4{x}^{2}+3x+12$.

izboknil3

Formula used:
The factors of a polynomial can be found 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 ${x}^{3}+4{x}^{2}+3x+12$.
This is a four term polynomial, factorization of this polynomial can be found by factor by grouping as,
${x}^{3}+4{x}^{2}+3x+12=\left({x}^{3}+4{x}^{2}\right)+\left(3x+12\right)$
$={x}^{2}\left(x+4\right)+3\left(x+4\right)$
As, $\left(x+4\right)$ is the common factor of the polynomial,
The polynomial can be factorized as,
${x}^{3}+4{x}^{2}+3x+12={x}^{2}\left(x+4\right)+3\left(x+4\right)$
$=\left(x+4\right)\left({x}^{2}+3\right)$
Therefore, the factorization of the polynomial ${x}^{3}+4{x}^{2}+3x+12$ is $\left(x+4\right)\left({x}^{2}+3\right)$.

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