Solve (x-1)^3(x^2-x-1)(2x+3)

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

Evaluate

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

Solution Steps

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

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

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

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

Use the distributive property to multiply x^{5}-4x^{4}+5x^{3}-2x-x^{2}+1 by 2x+3 and combine like terms.

$$2x^{6}-5x^{5}-2x^{4}+13x^{3}-7x^{2}-4x+3$$

Expand

$$2x^{6}-5x^{5}-2x^{4}+13x^{3}-7x^{2}-4x+3$$

Solution Steps

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

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

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

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

Use the distributive property to multiply x^{5}-4x^{4}+5x^{3}-2x-x^{2}+1 by 2x+3 and combine like terms.

$$2x^{6}-5x^{5}-2x^{4}+13x^{3}-7x^{2}-4x+3$$