Solve (xy+yz)^2-(xy-yz)^2

$$( x y + y z ) ^ { 2 } - ( x y - y z ) ^ { 2 }$$

Evaluate

$$4xzy^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(xy+yz\right)^{2}.

$$x^{2}y^{2}+2xyyz+y^{2}z^{2}-\left(xy-yz\right)^{2}$$

Multiply y and y to get y^{2}.

$$x^{2}y^{2}+2xy^{2}z+y^{2}z^{2}-\left(xy-yz\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(xy-yz\right)^{2}.

$$x^{2}y^{2}+2xy^{2}z+y^{2}z^{2}-\left(x^{2}y^{2}-2xyyz+y^{2}z^{2}\right)$$

Multiply y and y to get y^{2}.

$$x^{2}y^{2}+2xy^{2}z+y^{2}z^{2}-\left(x^{2}y^{2}-2xy^{2}z+y^{2}z^{2}\right)$$

To find the opposite of x^{2}y^{2}-2xy^{2}z+y^{2}z^{2}, find the opposite of each term.

$$x^{2}y^{2}+2xy^{2}z+y^{2}z^{2}-x^{2}y^{2}+2xy^{2}z-y^{2}z^{2}$$

Combine x^{2}y^{2} and -x^{2}y^{2} to get 0.

$$2xy^{2}z+y^{2}z^{2}+2xy^{2}z-y^{2}z^{2}$$

Combine 2xy^{2}z and 2xy^{2}z to get 4xy^{2}z.

$$4xy^{2}z+y^{2}z^{2}-y^{2}z^{2}$$

Combine y^{2}z^{2} and -y^{2}z^{2} to get 0.

$$4xy^{2}z$$

Expand

$$4xzy^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(xy+yz\right)^{2}.

$$x^{2}y^{2}+2xyyz+y^{2}z^{2}-\left(xy-yz\right)^{2}$$

Multiply y and y to get y^{2}.

$$x^{2}y^{2}+2xy^{2}z+y^{2}z^{2}-\left(xy-yz\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(xy-yz\right)^{2}.

$$x^{2}y^{2}+2xy^{2}z+y^{2}z^{2}-\left(x^{2}y^{2}-2xyyz+y^{2}z^{2}\right)$$

Multiply y and y to get y^{2}.

$$x^{2}y^{2}+2xy^{2}z+y^{2}z^{2}-\left(x^{2}y^{2}-2xy^{2}z+y^{2}z^{2}\right)$$

To find the opposite of x^{2}y^{2}-2xy^{2}z+y^{2}z^{2}, find the opposite of each term.

$$x^{2}y^{2}+2xy^{2}z+y^{2}z^{2}-x^{2}y^{2}+2xy^{2}z-y^{2}z^{2}$$

Combine x^{2}y^{2} and -x^{2}y^{2} to get 0.

$$2xy^{2}z+y^{2}z^{2}+2xy^{2}z-y^{2}z^{2}$$

Combine 2xy^{2}z and 2xy^{2}z to get 4xy^{2}z.

$$4xy^{2}z+y^{2}z^{2}-y^{2}z^{2}$$

Combine y^{2}z^{2} and -y^{2}z^{2} to get 0.

$$4xy^{2}z$$