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

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

Evaluate

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

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}.

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

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

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

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

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

Expand

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

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}.

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

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

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

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

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