Solve x^6+1/(x+1)^3(x^2+x+1)-(x-4)

$$\frac { x ^ { 6 } + 1 } { ( x + 1 ) ^ { 3 } ( x ^ { 2 } + x + 1 ) } - ( x - 4 )$$

Evaluate

$$\frac{9x^{4}+21x^{3}+24x^{2}+15x+5}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}$$

Short Solution Steps

Use the distributive property to multiply \left(x+1\right)^{3} by x^{2}+x+1.

$$\frac{x^{6}+1}{\left(x+1\right)^{3}x^{2}+\left(x+1\right)^{3}x+\left(x+1\right)^{3}}-\left(x-4\right)$$

To find the opposite of x-4, find the opposite of each term.

$$\frac{x^{6}+1}{\left(x+1\right)^{3}x^{2}+\left(x+1\right)^{3}x+\left(x+1\right)^{3}}-x+4$$

Factor \left(x+1\right)^{3}x^{2}+\left(x+1\right)^{3}x+\left(x+1\right)^{3}.

$$\frac{x^{6}+1}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}-x+4$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -x+4 times \frac{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}.

$$\frac{x^{6}+1}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}+\frac{\left(-x+4\right)\left(x^{2}+x+1\right)\left(x+1\right)^{3}}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}$$

Since \frac{x^{6}+1}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}} and \frac{\left(-x+4\right)\left(x^{2}+x+1\right)\left(x+1\right)^{3}}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}} have the same denominator, add them by adding their numerators.

$$\frac{x^{6}+1+\left(-x+4\right)\left(x^{2}+x+1\right)\left(x+1\right)^{3}}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}$$

Do the multiplications in x^{6}+1+\left(-x+4\right)\left(x^{2}+x+1\right)\left(x+1\right)^{3}.

$$\frac{x^{6}+1-x-x^{6}-4x^{5}-7x^{4}-7x^{3}-4x^{2}+4x^{5}+16x^{4}+28x^{3}+28x^{2}+16x+4}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}$$

Combine like terms in x^{6}+1-x-x^{6}-4x^{5}-7x^{4}-7x^{3}-4x^{2}+4x^{5}+16x^{4}+28x^{3}+28x^{2}+16x+4.

$$\frac{15x+5+9x^{4}+21x^{3}+24x^{2}}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}$$

Expand \left(x^{2}+x+1\right)\left(x+1\right)^{3}.

$$\frac{15x+5+9x^{4}+21x^{3}+24x^{2}}{x^{5}+4x^{4}+7x^{3}+7x^{2}+4x+1}$$

Expand

$$\frac{9x^{4}+21x^{3}+24x^{2}+15x+5}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}$$

Short Solution Steps

Use the distributive property to multiply \left(x+1\right)^{3} by x^{2}+x+1.

$$\frac{x^{6}+1}{\left(x+1\right)^{3}x^{2}+\left(x+1\right)^{3}x+\left(x+1\right)^{3}}-\left(x-4\right)$$

To find the opposite of x-4, find the opposite of each term.

$$\frac{x^{6}+1}{\left(x+1\right)^{3}x^{2}+\left(x+1\right)^{3}x+\left(x+1\right)^{3}}-x+4$$

Factor \left(x+1\right)^{3}x^{2}+\left(x+1\right)^{3}x+\left(x+1\right)^{3}.

$$\frac{x^{6}+1}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}-x+4$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -x+4 times \frac{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}.

$$\frac{x^{6}+1}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}+\frac{\left(-x+4\right)\left(x^{2}+x+1\right)\left(x+1\right)^{3}}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}$$

Since \frac{x^{6}+1}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}} and \frac{\left(-x+4\right)\left(x^{2}+x+1\right)\left(x+1\right)^{3}}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}} have the same denominator, add them by adding their numerators.

$$\frac{x^{6}+1+\left(-x+4\right)\left(x^{2}+x+1\right)\left(x+1\right)^{3}}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}$$

Do the multiplications in x^{6}+1+\left(-x+4\right)\left(x^{2}+x+1\right)\left(x+1\right)^{3}.

$$\frac{x^{6}+1-x-x^{6}-4x^{5}-7x^{4}-7x^{3}-4x^{2}+4x^{5}+16x^{4}+28x^{3}+28x^{2}+16x+4}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}$$

Combine like terms in x^{6}+1-x-x^{6}-4x^{5}-7x^{4}-7x^{3}-4x^{2}+4x^{5}+16x^{4}+28x^{3}+28x^{2}+16x+4.

$$\frac{15x+5+9x^{4}+21x^{3}+24x^{2}}{\left(x^{2}+x+1\right)\left(x+1\right)^{3}}$$

Expand \left(x^{2}+x+1\right)\left(x+1\right)^{3}.

$$\frac{15x+5+9x^{4}+21x^{3}+24x^{2}}{x^{5}+4x^{4}+7x^{3}+7x^{2}+4x+1}$$