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

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

Evaluate

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

Solution Steps

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

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

Apply the distributive property by multiplying each term of x^{2}+x by each term of x+2.

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

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

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

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

$$\frac{1}{x^{4}+3x^{3}+3x^{3}+9x^{2}+2x^{2}+6x}$$

Combine 3x^{3} and 3x^{3} to get 6x^{3}.

$$\frac{1}{x^{4}+6x^{3}+9x^{2}+2x^{2}+6x}$$

Combine 9x^{2} and 2x^{2} to get 11x^{2}.

$$\frac{1}{x^{4}+6x^{3}+11x^{2}+6x}$$

Expand

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

Solution Steps

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

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

Apply the distributive property by multiplying each term of x^{2}+x by each term of x+2.

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

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

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

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

$$\frac{1}{x^{4}+3x^{3}+3x^{3}+9x^{2}+2x^{2}+6x}$$

Combine 3x^{3} and 3x^{3} to get 6x^{3}.

$$\frac{1}{x^{4}+6x^{3}+9x^{2}+2x^{2}+6x}$$

Combine 9x^{2} and 2x^{2} to get 11x^{2}.

$$\frac{1}{x^{4}+6x^{3}+11x^{2}+6x}$$