Lipossig

2021-02-21

Need to calculate:The factorization of $2{z}^{3}+8{z}^{2}+5z+20$.

Clara Reese

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 $2{z}^{3}+8{z}^{2}+5z+20$.
This is a four term polynomial, factorization of this polynomial can be find by factor by grouping as,
$2{z}^{3}+8{z}^{2}+5z+20=\left(2{z}^{3}+8{z}^{2}\right)+\left(5z+20\right)$
$=2{z}^{2}\left(z+4\right)+5\left(z+4\right)$
As, $\left(z+4\right)$ is the common factor of the polynomial,
The polynomial can be factorized as,
$2{z}^{3}+8{z}^{2}+5z+20=2{z}^{2}\left(z+4\right)+5\left(z+4\right)$
$=\left(z+4\right)\left(2{z}^{2}+5\right)$
Therefore, the factorization of the polynomial is $\left(z+4\right)\left(2{z}^{2}+5\right)$.

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