Solve y^{3}+x^{2}y^{3}-y^{3}(x^{2}+1)+y^2(x^2+1)-y^2(x^2-1)

$$y ^ { 3 } + x ^ { 2 } y ^ { 3 } - y ^ { 3 } ( x ^ { 2 } + 1 ) + y ^ { 2 } ( x ^ { 2 } + 1 ) - y ^ { 2 } ( x ^ { 2 } - 1 )$$

Evaluate

$$2y^{2}$$

Solution Steps

Use the distributive property to multiply y^{3} by x^{2}+1.

$$y^{3}+x^{2}y^{3}-\left(y^{3}x^{2}+y^{3}\right)+y^{2}\left(x^{2}+1\right)-y^{2}\left(x^{2}-1\right)$$

Combine y^{3}+x^{2}y^{3} and -\left(y^{3}x^{2}+y^{3}\right) to get 0.

$$0+y^{2}\left(x^{2}+1\right)-y^{2}\left(x^{2}-1\right)$$

Use the distributive property to multiply y^{2} by x^{2}+1.

$$0+y^{2}x^{2}+y^{2}-y^{2}\left(x^{2}-1\right)$$

Anything plus zero gives itself.

$$y^{2}x^{2}+y^{2}-y^{2}\left(x^{2}-1\right)$$

Use the distributive property to multiply y^{2} by x^{2}-1.

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

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

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

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

$$y^{2}+y^{2}$$

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

$$2y^{2}$$

Expand

$$2y^{2}$$

Solution Steps

Use the distributive property to multiply y^{3} by x^{2}+1.

$$y^{3}+x^{2}y^{3}-\left(y^{3}x^{2}+y^{3}\right)+y^{2}\left(x^{2}+1\right)-y^{2}\left(x^{2}-1\right)$$

Combine y^{3}+x^{2}y^{3} and -\left(y^{3}x^{2}+y^{3}\right) to get 0.

$$0+y^{2}\left(x^{2}+1\right)-y^{2}\left(x^{2}-1\right)$$

Use the distributive property to multiply y^{2} by x^{2}+1.

$$0+y^{2}x^{2}+y^{2}-y^{2}\left(x^{2}-1\right)$$

Anything plus zero gives itself.

$$y^{2}x^{2}+y^{2}-y^{2}\left(x^{2}-1\right)$$

Use the distributive property to multiply y^{2} by x^{2}-1.

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

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

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

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

$$y^{2}+y^{2}$$

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

$$2y^{2}$$