Solve 6^{3}(x-1)^{2}(x+1)^3+3^2(x+1)^2(x-1)^3(3x)

$$6 ^ { 3 } ( x - 1 ) ^ { 2 } ( x + 1 ) ^ { 3 } + 3 ^ { 2 } ( x + 1 ) ^ { 2 } ( x - 1 ) ^ { 3 } ( 3 x )$$

Evaluate

$$27\left(x\left(x+7\right)+8\right)\left(x^{2}-1\right)^{2}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$6^{3}\left(x-1\right)^{2}\left(x+1\right)^{3}+3^{3}\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

Calculate 6 to the power of 3 and get 216.

$$216\left(x-1\right)^{2}\left(x+1\right)^{3}+3^{3}\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

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

$$216\left(x^{2}-2x+1\right)\left(x+1\right)^{3}+3^{3}\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

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

$$216\left(x^{2}-2x+1\right)\left(x^{3}+3x^{2}+3x+1\right)+3^{3}\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

Use the distributive property to multiply 216 by x^{2}-2x+1.

$$\left(216x^{2}-432x+216\right)\left(x^{3}+3x^{2}+3x+1\right)+3^{3}\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

Use the distributive property to multiply 216x^{2}-432x+216 by x^{3}+3x^{2}+3x+1 and combine like terms.

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+3^{3}\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

Calculate 3 to the power of 3 and get 27.

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+27\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

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

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+27\left(x^{2}+2x+1\right)\left(x-1\right)^{3}x$$

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

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+27\left(x^{2}+2x+1\right)\left(x^{3}-3x^{2}+3x-1\right)x$$

Use the distributive property to multiply 27 by x^{2}+2x+1.

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+\left(27x^{2}+54x+27\right)\left(x^{3}-3x^{2}+3x-1\right)x$$

Use the distributive property to multiply 27x^{2}+54x+27 by x^{3}-3x^{2}+3x-1 and combine like terms.

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+\left(27x^{5}-27x^{4}-54x^{3}+54x^{2}+27x-27\right)x$$

Use the distributive property to multiply 27x^{5}-27x^{4}-54x^{3}+54x^{2}+27x-27 by x.

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+27x^{6}-27x^{5}-54x^{4}+54x^{3}+27x^{2}-27x$$

Combine 216x^{5} and -27x^{5} to get 189x^{5}.

$$189x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+27x^{6}-54x^{4}+54x^{3}+27x^{2}-27x$$

Combine 216x^{4} and -54x^{4} to get 162x^{4}.

$$189x^{5}+162x^{4}-432x^{3}-432x^{2}+216x+216+27x^{6}+54x^{3}+27x^{2}-27x$$

Combine -432x^{3} and 54x^{3} to get -378x^{3}.

$$189x^{5}+162x^{4}-378x^{3}-432x^{2}+216x+216+27x^{6}+27x^{2}-27x$$

Combine -432x^{2} and 27x^{2} to get -405x^{2}.

$$189x^{5}+162x^{4}-378x^{3}-405x^{2}+216x+216+27x^{6}-27x$$

Combine 216x and -27x to get 189x.

$$189x^{5}+162x^{4}-378x^{3}-405x^{2}+189x+216+27x^{6}$$

Expand

$$27x^{6}+189x^{5}+162x^{4}-378x^{3}-405x^{2}+189x+216$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$6^{3}\left(x-1\right)^{2}\left(x+1\right)^{3}+3^{3}\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

Calculate 6 to the power of 3 and get 216.

$$216\left(x-1\right)^{2}\left(x+1\right)^{3}+3^{3}\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

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

$$216\left(x^{2}-2x+1\right)\left(x+1\right)^{3}+3^{3}\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

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

$$216\left(x^{2}-2x+1\right)\left(x^{3}+3x^{2}+3x+1\right)+3^{3}\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

Use the distributive property to multiply 216 by x^{2}-2x+1.

$$\left(216x^{2}-432x+216\right)\left(x^{3}+3x^{2}+3x+1\right)+3^{3}\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

Use the distributive property to multiply 216x^{2}-432x+216 by x^{3}+3x^{2}+3x+1 and combine like terms.

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+3^{3}\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

Calculate 3 to the power of 3 and get 27.

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+27\left(x+1\right)^{2}\left(x-1\right)^{3}x$$

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

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+27\left(x^{2}+2x+1\right)\left(x-1\right)^{3}x$$

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

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+27\left(x^{2}+2x+1\right)\left(x^{3}-3x^{2}+3x-1\right)x$$

Use the distributive property to multiply 27 by x^{2}+2x+1.

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+\left(27x^{2}+54x+27\right)\left(x^{3}-3x^{2}+3x-1\right)x$$

Use the distributive property to multiply 27x^{2}+54x+27 by x^{3}-3x^{2}+3x-1 and combine like terms.

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+\left(27x^{5}-27x^{4}-54x^{3}+54x^{2}+27x-27\right)x$$

Use the distributive property to multiply 27x^{5}-27x^{4}-54x^{3}+54x^{2}+27x-27 by x.

$$216x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+27x^{6}-27x^{5}-54x^{4}+54x^{3}+27x^{2}-27x$$

Combine 216x^{5} and -27x^{5} to get 189x^{5}.

$$189x^{5}+216x^{4}-432x^{3}-432x^{2}+216x+216+27x^{6}-54x^{4}+54x^{3}+27x^{2}-27x$$

Combine 216x^{4} and -54x^{4} to get 162x^{4}.

$$189x^{5}+162x^{4}-432x^{3}-432x^{2}+216x+216+27x^{6}+54x^{3}+27x^{2}-27x$$

Combine -432x^{3} and 54x^{3} to get -378x^{3}.

$$189x^{5}+162x^{4}-378x^{3}-432x^{2}+216x+216+27x^{6}+27x^{2}-27x$$

Combine -432x^{2} and 27x^{2} to get -405x^{2}.

$$189x^{5}+162x^{4}-378x^{3}-405x^{2}+216x+216+27x^{6}-27x$$

Combine 216x and -27x to get 189x.

$$189x^{5}+162x^{4}-378x^{3}-405x^{2}+189x+216+27x^{6}$$