Solve -20000+11800/1+x+13240/(1+x)^2+100/(1+x)^3

$$- 20000 + \frac { 11800 } { 1 + x } + \frac { 13240 } { ( 1 + x ) ^ { 2 } } + \frac { 100 } { ( 1 + x ) ^ { 3 } }$$

Evaluate

$$\frac{20\left(257-1158x-2410x^{2}-1000x^{3}\right)}{\left(x+1\right)^{3}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -20000 times \frac{1+x}{1+x}.

$$-\frac{20000\left(1+x\right)}{1+x}+\frac{11800}{1+x}+\frac{13240}{\left(1+x\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

Since -\frac{20000\left(1+x\right)}{1+x} and \frac{11800}{1+x} have the same denominator, add them by adding their numerators.

$$\frac{-20000\left(1+x\right)+11800}{1+x}+\frac{13240}{\left(1+x\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

Do the multiplications in -20000\left(1+x\right)+11800.

$$\frac{-20000-20000x+11800}{1+x}+\frac{13240}{\left(1+x\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

Combine like terms in -20000-20000x+11800.

$$\frac{-8200-20000x}{1+x}+\frac{13240}{\left(1+x\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1+x and \left(1+x\right)^{2} is \left(x+1\right)^{2}. Multiply \frac{-8200-20000x}{1+x} times \frac{x+1}{x+1}.

$$\frac{\left(-8200-20000x\right)\left(x+1\right)}{\left(x+1\right)^{2}}+\frac{13240}{\left(x+1\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

Since \frac{\left(-8200-20000x\right)\left(x+1\right)}{\left(x+1\right)^{2}} and \frac{13240}{\left(x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(-8200-20000x\right)\left(x+1\right)+13240}{\left(x+1\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

Do the multiplications in \left(-8200-20000x\right)\left(x+1\right)+13240.

$$\frac{-8200x-8200-20000x^{2}-20000x+13240}{\left(x+1\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

Combine like terms in -8200x-8200-20000x^{2}-20000x+13240.

$$\frac{-28200x+5040-20000x^{2}}{\left(x+1\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)^{2} and \left(1+x\right)^{3} is \left(x+1\right)^{3}. Multiply \frac{-28200x+5040-20000x^{2}}{\left(x+1\right)^{2}} times \frac{x+1}{x+1}.

$$\frac{\left(-28200x+5040-20000x^{2}\right)\left(x+1\right)}{\left(x+1\right)^{3}}+\frac{100}{\left(x+1\right)^{3}}$$

Since \frac{\left(-28200x+5040-20000x^{2}\right)\left(x+1\right)}{\left(x+1\right)^{3}} and \frac{100}{\left(x+1\right)^{3}} have the same denominator, add them by adding their numerators.

$$\frac{\left(-28200x+5040-20000x^{2}\right)\left(x+1\right)+100}{\left(x+1\right)^{3}}$$

Do the multiplications in \left(-28200x+5040-20000x^{2}\right)\left(x+1\right)+100.

$$\frac{-28200x^{2}-28200x+5040x+5040-20000x^{3}-20000x^{2}+100}{\left(x+1\right)^{3}}$$

Combine like terms in -28200x^{2}-28200x+5040x+5040-20000x^{3}-20000x^{2}+100.

$$\frac{-48200x^{2}-23160x+5140-20000x^{3}}{\left(x+1\right)^{3}}$$

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

$$\frac{-48200x^{2}-23160x+5140-20000x^{3}}{x^{3}+3x^{2}+3x+1}$$

Expand

$$-\frac{20\left(1000x^{3}+2410x^{2}+1158x-257\right)}{\left(x+1\right)^{3}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -20000 times \frac{1+x}{1+x}.

$$-\frac{20000\left(1+x\right)}{1+x}+\frac{11800}{1+x}+\frac{13240}{\left(1+x\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

Since -\frac{20000\left(1+x\right)}{1+x} and \frac{11800}{1+x} have the same denominator, add them by adding their numerators.

$$\frac{-20000\left(1+x\right)+11800}{1+x}+\frac{13240}{\left(1+x\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

Do the multiplications in -20000\left(1+x\right)+11800.

$$\frac{-20000-20000x+11800}{1+x}+\frac{13240}{\left(1+x\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

Combine like terms in -20000-20000x+11800.

$$\frac{-8200-20000x}{1+x}+\frac{13240}{\left(1+x\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1+x and \left(1+x\right)^{2} is \left(x+1\right)^{2}. Multiply \frac{-8200-20000x}{1+x} times \frac{x+1}{x+1}.

$$\frac{\left(-8200-20000x\right)\left(x+1\right)}{\left(x+1\right)^{2}}+\frac{13240}{\left(x+1\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

Since \frac{\left(-8200-20000x\right)\left(x+1\right)}{\left(x+1\right)^{2}} and \frac{13240}{\left(x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(-8200-20000x\right)\left(x+1\right)+13240}{\left(x+1\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

Do the multiplications in \left(-8200-20000x\right)\left(x+1\right)+13240.

$$\frac{-8200x-8200-20000x^{2}-20000x+13240}{\left(x+1\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

Combine like terms in -8200x-8200-20000x^{2}-20000x+13240.

$$\frac{-28200x+5040-20000x^{2}}{\left(x+1\right)^{2}}+\frac{100}{\left(1+x\right)^{3}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)^{2} and \left(1+x\right)^{3} is \left(x+1\right)^{3}. Multiply \frac{-28200x+5040-20000x^{2}}{\left(x+1\right)^{2}} times \frac{x+1}{x+1}.

$$\frac{\left(-28200x+5040-20000x^{2}\right)\left(x+1\right)}{\left(x+1\right)^{3}}+\frac{100}{\left(x+1\right)^{3}}$$

Since \frac{\left(-28200x+5040-20000x^{2}\right)\left(x+1\right)}{\left(x+1\right)^{3}} and \frac{100}{\left(x+1\right)^{3}} have the same denominator, add them by adding their numerators.

$$\frac{\left(-28200x+5040-20000x^{2}\right)\left(x+1\right)+100}{\left(x+1\right)^{3}}$$

Do the multiplications in \left(-28200x+5040-20000x^{2}\right)\left(x+1\right)+100.

$$\frac{-28200x^{2}-28200x+5040x+5040-20000x^{3}-20000x^{2}+100}{\left(x+1\right)^{3}}$$

Combine like terms in -28200x^{2}-28200x+5040x+5040-20000x^{3}-20000x^{2}+100.

$$\frac{-48200x^{2}-23160x+5140-20000x^{3}}{\left(x+1\right)^{3}}$$

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

$$\frac{-48200x^{2}-23160x+5140-20000x^{3}}{x^{3}+3x^{2}+3x+1}$$