Solve (y-3)(y^2+3y+9)-y(y-3)(y+3)-(y+3)^2

$$( y - 3 ) ( y ^ { 2 } + 3 y + 9 ) - y ( y - 3 ) ( y + 3 ) - ( y + 3 ) ^ { 2 }$$

Evaluate

$$-y^{2}+3y-36$$

Solution Steps

Use the distributive property to multiply y-3 by y^{2}+3y+9 and combine like terms.

$$y^{3}-27-y\left(y-3\right)\left(y+3\right)-\left(y+3\right)^{2}$$

Use the distributive property to multiply y by y-3.

$$y^{3}-27-\left(y^{2}-3y\right)\left(y+3\right)-\left(y+3\right)^{2}$$

Use the distributive property to multiply y^{2}-3y by y+3 and combine like terms.

$$y^{3}-27-\left(y^{3}-9y\right)-\left(y+3\right)^{2}$$

To find the opposite of y^{3}-9y, find the opposite of each term.

$$y^{3}-27-y^{3}+9y-\left(y+3\right)^{2}$$

Combine y^{3} and -y^{3} to get 0.

$$-27+9y-\left(y+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+3\right)^{2}.

$$-27+9y-\left(y^{2}+6y+9\right)$$

To find the opposite of y^{2}+6y+9, find the opposite of each term.

$$-27+9y-y^{2}-6y-9$$

Combine 9y and -6y to get 3y.

$$-27+3y-y^{2}-9$$

Subtract 9 from -27 to get -36.

$$-36+3y-y^{2}$$

Expand

$$-y^{2}+3y-36$$

Solution Steps

Use the distributive property to multiply y-3 by y^{2}+3y+9 and combine like terms.

$$y^{3}-27-y\left(y-3\right)\left(y+3\right)-\left(y+3\right)^{2}$$

Use the distributive property to multiply y by y-3.

$$y^{3}-27-\left(y^{2}-3y\right)\left(y+3\right)-\left(y+3\right)^{2}$$

Use the distributive property to multiply y^{2}-3y by y+3 and combine like terms.

$$y^{3}-27-\left(y^{3}-9y\right)-\left(y+3\right)^{2}$$

To find the opposite of y^{3}-9y, find the opposite of each term.

$$y^{3}-27-y^{3}+9y-\left(y+3\right)^{2}$$

Combine y^{3} and -y^{3} to get 0.

$$-27+9y-\left(y+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+3\right)^{2}.

$$-27+9y-\left(y^{2}+6y+9\right)$$

To find the opposite of y^{2}+6y+9, find the opposite of each term.

$$-27+9y-y^{2}-6y-9$$

Combine 9y and -6y to get 3y.

$$-27+3y-y^{2}-9$$

Subtract 9 from -27 to get -36.

$$-36+3y-y^{2}$$