Solve y=-(x^2+1/2)(x-1)^2(x+1)(x-2)

$$y = - ( x ^ { 2 } + \frac { 1 } { 2 } ) ( x - 1 ) ^ { 2 } ( x + 1 ) ( x - 2 )$$

Solve for y

$$y=-\frac{\left(x-2\right)\left(x+1\right)\left(x-1\right)^{2}\left(2x^{2}+1\right)}{2}$$

Solution Steps

To find the opposite of x^{2}+\frac{1}{2}, find the opposite of each term.

$$y=\left(-x^{2}-\frac{1}{2}\right)\left(x-1\right)^{2}\left(x+1\right)\left(x-2\right)$$

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

$$y=\left(-x^{2}-\frac{1}{2}\right)\left(x^{2}-2x+1\right)\left(x+1\right)\left(x-2\right)$$

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

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

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

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

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

$$y=-x^{6}+3x^{5}-\frac{3}{2}x^{4}-\frac{3}{2}x^{3}+\frac{3}{2}x^{2}-\frac{3}{2}x+1$$