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

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

Evaluate

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

Solution Steps

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

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

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

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

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

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

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

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

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

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

Apply the distributive property by multiplying each term of x^{3}-\frac{5}{2}x^{2}+x\times \frac{3}{2} by each term of x+\frac{1}{2}.

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

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

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

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

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

Combine x^{3}\times \frac{1}{2} and -\frac{5}{2}x^{3} to get -2x^{3}.

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

Multiply -\frac{5}{2} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{-5}{2\times 2}.

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

Fraction \frac{-5}{4} can be rewritten as -\frac{5}{4} by extracting the negative sign.

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

Combine -\frac{5}{4}x^{2} and x^{2}\times \frac{3}{2} to get \frac{1}{4}x^{2}.

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

Multiply \frac{3}{2} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{3\times 1}{2\times 2}.

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

Expand

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

Solution Steps

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

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

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

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

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

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

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

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

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

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

Apply the distributive property by multiplying each term of x^{3}-\frac{5}{2}x^{2}+x\times \frac{3}{2} by each term of x+\frac{1}{2}.

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

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

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

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

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

Combine x^{3}\times \frac{1}{2} and -\frac{5}{2}x^{3} to get -2x^{3}.

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

Multiply -\frac{5}{2} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{-5}{2\times 2}.

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

Fraction \frac{-5}{4} can be rewritten as -\frac{5}{4} by extracting the negative sign.

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

Combine -\frac{5}{4}x^{2} and x^{2}\times \frac{3}{2} to get \frac{1}{4}x^{2}.

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

Multiply \frac{3}{2} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{3\times 1}{2\times 2}.

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