Solve -(x^2-3/2x+2)(-x+1/3)+x(x+1)(x-1)-33/2(1/3x+1)^2

$$- ( x ^ { 2 } - \frac { 3 } { 2 } x + 2 ) ( - x + \frac { 1 } { 3 } ) + x ( x + 1 ) ( x - 1 ) - \frac { 33 } { 2 } ( \frac { 1 } { 3 } x + 1 ) ^ { 2 }$$

Evaluate

$$2x^{3}-\frac{11x^{2}}{3}-\frac{19x}{2}-\frac{103}{6}$$

Solution Steps

To find the opposite of x^{2}-\frac{3}{2}x+2, find the opposite of each term.

$$\left(-x^{2}+\frac{3}{2}x-2\right)\left(-x+\frac{1}{3}\right)+x\left(x+1\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

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

$$-x^{2}\left(-x\right)-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{1}{2}x-2\left(-x\right)-\frac{2}{3}+x\left(x+1\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

Multiply -1 and -1 to get 1.

$$x^{2}x-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{1}{2}x-2\left(-x\right)-\frac{2}{3}+x\left(x+1\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{1}{2}x-2\left(-x\right)-\frac{2}{3}+x\left(x+1\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

Multiply -2 and -1 to get 2.

$$x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{1}{2}x+2x-\frac{2}{3}+x\left(x+1\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

Combine \frac{1}{2}x and 2x to get \frac{5}{2}x.

$$x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{5}{2}x-\frac{2}{3}+x\left(x+1\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

Use the distributive property to multiply x by x+1.

$$x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{5}{2}x-\frac{2}{3}+\left(x^{2}+x\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

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

$$x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{5}{2}x-\frac{2}{3}+x^{3}-x-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

Combine x^{3} and x^{3} to get 2x^{3}.

$$2x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{5}{2}x-\frac{2}{3}-x-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

Combine \frac{5}{2}x and -x to get \frac{3}{2}x.

$$2x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{3}{2}x-\frac{2}{3}-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

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

$$2x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{3}{2}x-\frac{2}{3}-\frac{33}{2}\left(\frac{1}{9}x^{2}+\frac{2}{3}x+1\right)$$

Use the distributive property to multiply -\frac{33}{2} by \frac{1}{9}x^{2}+\frac{2}{3}x+1.

$$2x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{3}{2}x-\frac{2}{3}-\frac{11}{6}x^{2}-11x-\frac{33}{2}$$

Combine -\frac{1}{3}x^{2} and -\frac{11}{6}x^{2} to get -\frac{13}{6}x^{2}.

$$2x^{3}-\frac{13}{6}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{3}{2}x-\frac{2}{3}-11x-\frac{33}{2}$$

Combine \frac{3}{2}x and -11x to get -\frac{19}{2}x.

$$2x^{3}-\frac{13}{6}x^{2}+\frac{3}{2}x\left(-x\right)-\frac{19}{2}x-\frac{2}{3}-\frac{33}{2}$$

Subtract \frac{33}{2} from -\frac{2}{3} to get -\frac{103}{6}.

$$2x^{3}-\frac{13}{6}x^{2}+\frac{3}{2}x\left(-x\right)-\frac{19}{2}x-\frac{103}{6}$$

Multiply x and x to get x^{2}.

$$2x^{3}-\frac{13}{6}x^{2}+\frac{3}{2}x^{2}\left(-1\right)-\frac{19}{2}x-\frac{103}{6}$$

Multiply \frac{3}{2} and -1 to get -\frac{3}{2}.

$$2x^{3}-\frac{13}{6}x^{2}-\frac{3}{2}x^{2}-\frac{19}{2}x-\frac{103}{6}$$

Combine -\frac{13}{6}x^{2} and -\frac{3}{2}x^{2} to get -\frac{11}{3}x^{2}.

$$2x^{3}-\frac{11}{3}x^{2}-\frac{19}{2}x-\frac{103}{6}$$

Expand

$$2x^{3}-\frac{11x^{2}}{3}-\frac{19x}{2}-\frac{103}{6}$$

Solution Steps

To find the opposite of x^{2}-\frac{3}{2}x+2, find the opposite of each term.

$$\left(-x^{2}+\frac{3}{2}x-2\right)\left(-x+\frac{1}{3}\right)+x\left(x+1\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

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

$$-x^{2}\left(-x\right)-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{1}{2}x-2\left(-x\right)-\frac{2}{3}+x\left(x+1\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

Multiply -1 and -1 to get 1.

$$x^{2}x-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{1}{2}x-2\left(-x\right)-\frac{2}{3}+x\left(x+1\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{1}{2}x-2\left(-x\right)-\frac{2}{3}+x\left(x+1\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

Multiply -2 and -1 to get 2.

$$x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{1}{2}x+2x-\frac{2}{3}+x\left(x+1\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

Combine \frac{1}{2}x and 2x to get \frac{5}{2}x.

$$x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{5}{2}x-\frac{2}{3}+x\left(x+1\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

Use the distributive property to multiply x by x+1.

$$x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{5}{2}x-\frac{2}{3}+\left(x^{2}+x\right)\left(x-1\right)-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

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

$$x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{5}{2}x-\frac{2}{3}+x^{3}-x-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

Combine x^{3} and x^{3} to get 2x^{3}.

$$2x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{5}{2}x-\frac{2}{3}-x-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

Combine \frac{5}{2}x and -x to get \frac{3}{2}x.

$$2x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{3}{2}x-\frac{2}{3}-\frac{33}{2}\left(\frac{1}{3}x+1\right)^{2}$$

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

$$2x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{3}{2}x-\frac{2}{3}-\frac{33}{2}\left(\frac{1}{9}x^{2}+\frac{2}{3}x+1\right)$$

Use the distributive property to multiply -\frac{33}{2} by \frac{1}{9}x^{2}+\frac{2}{3}x+1.

$$2x^{3}-\frac{1}{3}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{3}{2}x-\frac{2}{3}-\frac{11}{6}x^{2}-11x-\frac{33}{2}$$

Combine -\frac{1}{3}x^{2} and -\frac{11}{6}x^{2} to get -\frac{13}{6}x^{2}.

$$2x^{3}-\frac{13}{6}x^{2}+\frac{3}{2}x\left(-x\right)+\frac{3}{2}x-\frac{2}{3}-11x-\frac{33}{2}$$

Combine \frac{3}{2}x and -11x to get -\frac{19}{2}x.

$$2x^{3}-\frac{13}{6}x^{2}+\frac{3}{2}x\left(-x\right)-\frac{19}{2}x-\frac{2}{3}-\frac{33}{2}$$

Subtract \frac{33}{2} from -\frac{2}{3} to get -\frac{103}{6}.

$$2x^{3}-\frac{13}{6}x^{2}+\frac{3}{2}x\left(-x\right)-\frac{19}{2}x-\frac{103}{6}$$

Multiply x and x to get x^{2}.

$$2x^{3}-\frac{13}{6}x^{2}+\frac{3}{2}x^{2}\left(-1\right)-\frac{19}{2}x-\frac{103}{6}$$

Multiply \frac{3}{2} and -1 to get -\frac{3}{2}.

$$2x^{3}-\frac{13}{6}x^{2}-\frac{3}{2}x^{2}-\frac{19}{2}x-\frac{103}{6}$$

Combine -\frac{13}{6}x^{2} and -\frac{3}{2}x^{2} to get -\frac{11}{3}x^{2}.

$$2x^{3}-\frac{11}{3}x^{2}-\frac{19}{2}x-\frac{103}{6}$$