Solve (3x-2y)^{3}-2(4y+3y)^2-3x(9x^2-18xy+12y^2)-2y^3(-4y-49)

$$( 3 x - 2 y ) ^ { 3 } - 2 ( 4 y + 3 y ) ^ { 2 } - 3 x ( 9 x ^ { 2 } - 18 x y + 12 y ^ { 2 } ) - 2 y ^ { 3 } ( - 4 y - 49 )$$

Evaluate

$$2\left(y-1\right)\left(4y+49\right)y^{2}$$

Solution Steps

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

$$27x^{3}-54x^{2}y+36xy^{2}-8y^{3}-2\left(4y+3y\right)^{2}-3x\left(9x^{2}-18xy+12y^{2}\right)-2y^{3}\left(-4y-49\right)$$

Combine 4y and 3y to get 7y.

$$27x^{3}-54x^{2}y+36xy^{2}-8y^{3}-2\times \left(7y\right)^{2}-3x\left(9x^{2}-18xy+12y^{2}\right)-2y^{3}\left(-4y-49\right)$$

Expand \left(7y\right)^{2}.

$$27x^{3}-54x^{2}y+36xy^{2}-8y^{3}-2\times 7^{2}y^{2}-3x\left(9x^{2}-18xy+12y^{2}\right)-2y^{3}\left(-4y-49\right)$$

Calculate 7 to the power of 2 and get 49.

$$27x^{3}-54x^{2}y+36xy^{2}-8y^{3}-2\times 49y^{2}-3x\left(9x^{2}-18xy+12y^{2}\right)-2y^{3}\left(-4y-49\right)$$

Multiply 2 and 49 to get 98.

$$27x^{3}-54x^{2}y+36xy^{2}-8y^{3}-98y^{2}-3x\left(9x^{2}-18xy+12y^{2}\right)-2y^{3}\left(-4y-49\right)$$

Use the distributive property to multiply -3x by 9x^{2}-18xy+12y^{2}.

$$27x^{3}-54x^{2}y+36xy^{2}-8y^{3}-98y^{2}-27x^{3}+54x^{2}y-36xy^{2}-2y^{3}\left(-4y-49\right)$$

Combine 27x^{3} and -27x^{3} to get 0.

$$-54x^{2}y+36xy^{2}-8y^{3}-98y^{2}+54x^{2}y-36xy^{2}-2y^{3}\left(-4y-49\right)$$

Combine -54x^{2}y and 54x^{2}y to get 0.

$$36xy^{2}-8y^{3}-98y^{2}-36xy^{2}-2y^{3}\left(-4y-49\right)$$

Combine 36xy^{2} and -36xy^{2} to get 0.

$$-8y^{3}-98y^{2}-2y^{3}\left(-4y-49\right)$$

Use the distributive property to multiply -2y^{3} by -4y-49.

$$-8y^{3}-98y^{2}+8y^{4}+98y^{3}$$

Combine -8y^{3} and 98y^{3} to get 90y^{3}.

$$90y^{3}-98y^{2}+8y^{4}$$

Expand

$$8y^{4}+90y^{3}-98y^{2}$$

Solution Steps

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

$$27x^{3}-54x^{2}y+36xy^{2}-8y^{3}-2\left(4y+3y\right)^{2}-3x\left(9x^{2}-18xy+12y^{2}\right)-2y^{3}\left(-4y-49\right)$$

Combine 4y and 3y to get 7y.

$$27x^{3}-54x^{2}y+36xy^{2}-8y^{3}-2\times \left(7y\right)^{2}-3x\left(9x^{2}-18xy+12y^{2}\right)-2y^{3}\left(-4y-49\right)$$

Expand \left(7y\right)^{2}.

$$27x^{3}-54x^{2}y+36xy^{2}-8y^{3}-2\times 7^{2}y^{2}-3x\left(9x^{2}-18xy+12y^{2}\right)-2y^{3}\left(-4y-49\right)$$

Calculate 7 to the power of 2 and get 49.

$$27x^{3}-54x^{2}y+36xy^{2}-8y^{3}-2\times 49y^{2}-3x\left(9x^{2}-18xy+12y^{2}\right)-2y^{3}\left(-4y-49\right)$$

Multiply 2 and 49 to get 98.

$$27x^{3}-54x^{2}y+36xy^{2}-8y^{3}-98y^{2}-3x\left(9x^{2}-18xy+12y^{2}\right)-2y^{3}\left(-4y-49\right)$$

Use the distributive property to multiply -3x by 9x^{2}-18xy+12y^{2}.

$$27x^{3}-54x^{2}y+36xy^{2}-8y^{3}-98y^{2}-27x^{3}+54x^{2}y-36xy^{2}-2y^{3}\left(-4y-49\right)$$

Combine 27x^{3} and -27x^{3} to get 0.

$$-54x^{2}y+36xy^{2}-8y^{3}-98y^{2}+54x^{2}y-36xy^{2}-2y^{3}\left(-4y-49\right)$$

Combine -54x^{2}y and 54x^{2}y to get 0.

$$36xy^{2}-8y^{3}-98y^{2}-36xy^{2}-2y^{3}\left(-4y-49\right)$$

Combine 36xy^{2} and -36xy^{2} to get 0.

$$-8y^{3}-98y^{2}-2y^{3}\left(-4y-49\right)$$

Use the distributive property to multiply -2y^{3} by -4y-49.

$$-8y^{3}-98y^{2}+8y^{4}+98y^{3}$$

Combine -8y^{3} and 98y^{3} to get 90y^{3}.

$$90y^{3}-98y^{2}+8y^{4}$$