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

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

Evaluate

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

Solution Steps

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

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

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

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

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

$$x^{5}-x^{4}-5x^{3}+x^{2}+8x+4$$

Expand

$$x^{5}-x^{4}-5x^{3}+x^{2}+8x+4$$

Solution Steps

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

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

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

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

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

$$x^{5}-x^{4}-5x^{3}+x^{2}+8x+4$$