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

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

Evaluate

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

Solution Steps

Apply the distributive property by multiplying each term of x-3 by each term of x-1.

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

Combine -x and -3x to get -4x.

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

Apply the distributive property by multiplying each term of x^{2}-4x+3 by each term of x+\frac{1}{2}.

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

Combine x^{2}\times \frac{1}{2} and -4x^{2} to get -\frac{7}{2}x^{2}.

$$\left(x^{3}-\frac{7}{2}x^{2}-4x\times \frac{1}{2}+3x+3\times \frac{1}{2}\right)\left(x+3\right)$$

Multiply -4 and \frac{1}{2} to get \frac{-4}{2}.

$$\left(x^{3}-\frac{7}{2}x^{2}+\frac{-4}{2}x+3x+3\times \frac{1}{2}\right)\left(x+3\right)$$

Divide -4 by 2 to get -2.

$$\left(x^{3}-\frac{7}{2}x^{2}-2x+3x+3\times \frac{1}{2}\right)\left(x+3\right)$$

Combine -2x and 3x to get x.

$$\left(x^{3}-\frac{7}{2}x^{2}+x+3\times \frac{1}{2}\right)\left(x+3\right)$$

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

$$\left(x^{3}-\frac{7}{2}x^{2}+x+\frac{3}{2}\right)\left(x+3\right)$$

Apply the distributive property by multiplying each term of x^{3}-\frac{7}{2}x^{2}+x+\frac{3}{2} by each term of x+3.

$$x^{4}+3x^{3}-\frac{7}{2}x^{2}x-\frac{7}{2}x^{2}\times 3+x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

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

$$x^{4}+3x^{3}-\frac{7}{2}x^{3}-\frac{7}{2}x^{2}\times 3+x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

Combine 3x^{3} and -\frac{7}{2}x^{3} to get -\frac{1}{2}x^{3}.

$$x^{4}-\frac{1}{2}x^{3}-\frac{7}{2}x^{2}\times 3+x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

Express -\frac{7}{2}\times 3 as a single fraction.

$$x^{4}-\frac{1}{2}x^{3}+\frac{-7\times 3}{2}x^{2}+x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

Multiply -7 and 3 to get -21.

$$x^{4}-\frac{1}{2}x^{3}+\frac{-21}{2}x^{2}+x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

Fraction \frac{-21}{2} can be rewritten as -\frac{21}{2} by extracting the negative sign.

$$x^{4}-\frac{1}{2}x^{3}-\frac{21}{2}x^{2}+x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

Combine -\frac{21}{2}x^{2} and x^{2} to get -\frac{19}{2}x^{2}.

$$x^{4}-\frac{1}{2}x^{3}-\frac{19}{2}x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

Combine 3x and \frac{3}{2}x to get \frac{9}{2}x.

$$x^{4}-\frac{1}{2}x^{3}-\frac{19}{2}x^{2}+\frac{9}{2}x+\frac{3}{2}\times 3$$

Express \frac{3}{2}\times 3 as a single fraction.

$$x^{4}-\frac{1}{2}x^{3}-\frac{19}{2}x^{2}+\frac{9}{2}x+\frac{3\times 3}{2}$$

Multiply 3 and 3 to get 9.

$$x^{4}-\frac{1}{2}x^{3}-\frac{19}{2}x^{2}+\frac{9}{2}x+\frac{9}{2}$$

Expand

$$x^{4}-\frac{x^{3}}{2}-\frac{19x^{2}}{2}+\frac{9x}{2}+\frac{9}{2}$$

Solution Steps

Apply the distributive property by multiplying each term of x-3 by each term of x-1.

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

Combine -x and -3x to get -4x.

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

Apply the distributive property by multiplying each term of x^{2}-4x+3 by each term of x+\frac{1}{2}.

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

Combine x^{2}\times \frac{1}{2} and -4x^{2} to get -\frac{7}{2}x^{2}.

$$\left(x^{3}-\frac{7}{2}x^{2}-4x\times \frac{1}{2}+3x+3\times \frac{1}{2}\right)\left(x+3\right)$$

Multiply -4 and \frac{1}{2} to get \frac{-4}{2}.

$$\left(x^{3}-\frac{7}{2}x^{2}+\frac{-4}{2}x+3x+3\times \frac{1}{2}\right)\left(x+3\right)$$

Divide -4 by 2 to get -2.

$$\left(x^{3}-\frac{7}{2}x^{2}-2x+3x+3\times \frac{1}{2}\right)\left(x+3\right)$$

Combine -2x and 3x to get x.

$$\left(x^{3}-\frac{7}{2}x^{2}+x+3\times \frac{1}{2}\right)\left(x+3\right)$$

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

$$\left(x^{3}-\frac{7}{2}x^{2}+x+\frac{3}{2}\right)\left(x+3\right)$$

Apply the distributive property by multiplying each term of x^{3}-\frac{7}{2}x^{2}+x+\frac{3}{2} by each term of x+3.

$$x^{4}+3x^{3}-\frac{7}{2}x^{2}x-\frac{7}{2}x^{2}\times 3+x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

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

$$x^{4}+3x^{3}-\frac{7}{2}x^{3}-\frac{7}{2}x^{2}\times 3+x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

Combine 3x^{3} and -\frac{7}{2}x^{3} to get -\frac{1}{2}x^{3}.

$$x^{4}-\frac{1}{2}x^{3}-\frac{7}{2}x^{2}\times 3+x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

Express -\frac{7}{2}\times 3 as a single fraction.

$$x^{4}-\frac{1}{2}x^{3}+\frac{-7\times 3}{2}x^{2}+x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

Multiply -7 and 3 to get -21.

$$x^{4}-\frac{1}{2}x^{3}+\frac{-21}{2}x^{2}+x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

Fraction \frac{-21}{2} can be rewritten as -\frac{21}{2} by extracting the negative sign.

$$x^{4}-\frac{1}{2}x^{3}-\frac{21}{2}x^{2}+x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

Combine -\frac{21}{2}x^{2} and x^{2} to get -\frac{19}{2}x^{2}.

$$x^{4}-\frac{1}{2}x^{3}-\frac{19}{2}x^{2}+3x+\frac{3}{2}x+\frac{3}{2}\times 3$$

Combine 3x and \frac{3}{2}x to get \frac{9}{2}x.

$$x^{4}-\frac{1}{2}x^{3}-\frac{19}{2}x^{2}+\frac{9}{2}x+\frac{3}{2}\times 3$$

Express \frac{3}{2}\times 3 as a single fraction.

$$x^{4}-\frac{1}{2}x^{3}-\frac{19}{2}x^{2}+\frac{9}{2}x+\frac{3\times 3}{2}$$

Multiply 3 and 3 to get 9.

$$x^{4}-\frac{1}{2}x^{3}-\frac{19}{2}x^{2}+\frac{9}{2}x+\frac{9}{2}$$