# Factor each polynomial. 6x^{3}+3x^{2}y+2xz^{2}+yz^{2}

Factor each polynomial. $6{x}^{3}+3{x}^{2}y+2x{z}^{2}+y{z}^{2}$
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Ella Williams
Step 1
To factor the polynomial $6{x}^{3}+3{x}^{2}y+2x{z}^{2}+y{z}^{2}$, first take the common factor $3{x}^{2}$ from the first two terms and ${z}^{2}$ from the last two terms, and then take out the common term (2x+y) from the resulting expression to get the factorized form of the equation, that is;
$6{x}^{3}+3{x}^{2}y+2x{z}^{2}+y{z}^{2}=3{x}^{2}\cdot 2x+3{x}^{2}y+2x{z}^{2}+y{z}^{2}$
$=3{x}^{2}\left(2x+y\right)+{z}^{2}\left(2x+y\right)$
$=\left(2x+y\right)\left(3{x}^{2}+{z}^{2}\right)$
Step 2
Now to check the obtained result, expand the expression by cross multiplication and check whether it forms the initial polynomial, as follows;
$\left(2x+y\right)\left(3{x}^{2}+{z}^{2}\right)=2x\left(3{x}^{2}\right)+2x\left({z}^{2}\right)+y\left(3{x}^{2}\right)+y\left({z}^{2}\right)$
$=6{x}^{3}+2x{z}^{2}+3y{x}^{2}+y{z}^{2}$
Therefore the factorized form of the polynomial $6{x}^{3}+3{x}^{2}y+2x{z}^{2}+y{z}^{2}$ is $\left(2x+y\right)\left(3{x}^{2}+{z}^{2}\right)$