Solve left(y+2right)^3+8left(1-yright)^3

$${ \left(y+2 \right) }^{ 3 } +8 { \left(1-y \right) }^{ 3 }$$

Evaluate

$$\left(4-y\right)\left(7y^{2}-2y+4\right)$$

Solution Steps

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

$$y^{3}+6y^{2}+12y+8+8\left(1-y\right)^{3}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(1-y\right)^{3}.

$$y^{3}+6y^{2}+12y+8+8\left(1-3y+3y^{2}-y^{3}\right)$$

Use the distributive property to multiply 8 by 1-3y+3y^{2}-y^{3}.

$$y^{3}+6y^{2}+12y+8+8-24y+24y^{2}-8y^{3}$$

Add 8 and 8 to get 16.

$$y^{3}+6y^{2}+12y+16-24y+24y^{2}-8y^{3}$$

Combine 12y and -24y to get -12y.

$$y^{3}+6y^{2}-12y+16+24y^{2}-8y^{3}$$

Combine 6y^{2} and 24y^{2} to get 30y^{2}.

$$y^{3}+30y^{2}-12y+16-8y^{3}$$

Combine y^{3} and -8y^{3} to get -7y^{3}.

$$-7y^{3}+30y^{2}-12y+16$$

Expand

$$16-12y+30y^{2}-7y^{3}$$

Solution Steps

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

$$y^{3}+6y^{2}+12y+8+8\left(1-y\right)^{3}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(1-y\right)^{3}.

$$y^{3}+6y^{2}+12y+8+8\left(1-3y+3y^{2}-y^{3}\right)$$

Use the distributive property to multiply 8 by 1-3y+3y^{2}-y^{3}.

$$y^{3}+6y^{2}+12y+8+8-24y+24y^{2}-8y^{3}$$

Add 8 and 8 to get 16.

$$y^{3}+6y^{2}+12y+16-24y+24y^{2}-8y^{3}$$

Combine 12y and -24y to get -12y.

$$y^{3}+6y^{2}-12y+16+24y^{2}-8y^{3}$$

Combine 6y^{2} and 24y^{2} to get 30y^{2}.

$$y^{3}+30y^{2}-12y+16-8y^{3}$$

Combine y^{3} and -8y^{3} to get -7y^{3}.

$$-7y^{3}+30y^{2}-12y+16$$