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

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

Evaluate

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

Solution Steps

To find the opposite of x-2, find the opposite of each term.

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

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

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

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

$$\left(-x+2\right)\left(x^{2}+2x+1\right)\left(x^{2}-6x+9\right)$$

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

$$\left(-x^{3}+3x+2\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply -x^{3}+3x+2 by x^{2}-6x+9 and combine like terms.

$$-x^{5}+6x^{4}-6x^{3}-16x^{2}+15x+18$$

Expand

$$18+15x-16x^{2}-6x^{3}+6x^{4}-x^{5}$$

Solution Steps

To find the opposite of x-2, find the opposite of each term.

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

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

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

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

$$\left(-x+2\right)\left(x^{2}+2x+1\right)\left(x^{2}-6x+9\right)$$

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

$$\left(-x^{3}+3x+2\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply -x^{3}+3x+2 by x^{2}-6x+9 and combine like terms.

$$-x^{5}+6x^{4}-6x^{3}-16x^{2}+15x+18$$