Solve 1*(x-1)*(x+1)*(x-1/2)*(x-2)

$$1 \times (x-1) \times (x+1) \times (x- \frac{ 1 }{ 2 } ) \times (x-2)$$

Evaluate

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

Solution Steps

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

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

Apply the distributive property by multiplying each term of x-1 by each term of x+1.

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

Combine x and -x to get 0.

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

Apply the distributive property by multiplying each term of x^{2}-1 by each term of x-\frac{1}{2}.

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

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

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

Apply the distributive property by multiplying each term of x^{3}+x^{2}\left(-\frac{1}{2}\right)-x+\frac{1}{2} by each term of x-2.

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

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

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

Combine -2x^{3} and x^{3}\left(-\frac{1}{2}\right) to get -\frac{5}{2}x^{3}.

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

Express -\frac{1}{2}\left(-2\right) as a single fraction.

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

Multiply -1 and -2 to get 2.

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

Divide 2 by 2 to get 1.

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

Combine x^{2}\times 1 and -x^{2} to get 0.

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

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

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

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

$$x^{4}-\frac{5}{2}x^{3}+\frac{5}{2}x+\frac{-2}{2}$$

Divide -2 by 2 to get -1.

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

Expand

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

Solution Steps

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

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

Apply the distributive property by multiplying each term of x-1 by each term of x+1.

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

Combine x and -x to get 0.

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

Apply the distributive property by multiplying each term of x^{2}-1 by each term of x-\frac{1}{2}.

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

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

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

Apply the distributive property by multiplying each term of x^{3}+x^{2}\left(-\frac{1}{2}\right)-x+\frac{1}{2} by each term of x-2.

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

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

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

Combine -2x^{3} and x^{3}\left(-\frac{1}{2}\right) to get -\frac{5}{2}x^{3}.

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

Express -\frac{1}{2}\left(-2\right) as a single fraction.

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

Multiply -1 and -2 to get 2.

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

Divide 2 by 2 to get 1.

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

Combine x^{2}\times 1 and -x^{2} to get 0.

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

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

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

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

$$x^{4}-\frac{5}{2}x^{3}+\frac{5}{2}x+\frac{-2}{2}$$

Divide -2 by 2 to get -1.

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