Solve 1/p-3-1/(p-1)^2+1/p+1

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

Evaluate

$$\frac{2p^{3}-7p^{2}+8p+1}{\left(p-3\right)\left(p+1\right)\left(p-1\right)^{2}}$$

Short Solution Steps

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

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

Since \frac{\left(p-1\right)^{2}}{\left(p-3\right)\left(p-1\right)^{2}} and \frac{p-3}{\left(p-3\right)\left(p-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(p-1\right)^{2}-\left(p-3\right).

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

Combine like terms in p^{2}-2p+1-p+3.

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

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

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

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

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

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

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

Combine like terms in p^{3}+p^{2}-3p^{2}-3p+4p+4+p^{3}-2p^{2}+p-3p^{2}+6p-3.

$$\frac{2p^{3}-7p^{2}+8p+1}{\left(p-3\right)\left(p+1\right)\left(p-1\right)^{2}}$$

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

$$\frac{2p^{3}-7p^{2}+8p+1}{p^{4}-4p^{3}+2p^{2}+4p-3}$$

Expand

$$\frac{2p^{3}-7p^{2}+8p+1}{\left(p-3\right)\left(p+1\right)\left(p-1\right)^{2}}$$

Short Solution Steps

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

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

Since \frac{\left(p-1\right)^{2}}{\left(p-3\right)\left(p-1\right)^{2}} and \frac{p-3}{\left(p-3\right)\left(p-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(p-1\right)^{2}-\left(p-3\right).

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

Combine like terms in p^{2}-2p+1-p+3.

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

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

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

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

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

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

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

Combine like terms in p^{3}+p^{2}-3p^{2}-3p+4p+4+p^{3}-2p^{2}+p-3p^{2}+6p-3.

$$\frac{2p^{3}-7p^{2}+8p+1}{\left(p-3\right)\left(p+1\right)\left(p-1\right)^{2}}$$

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

$$\frac{2p^{3}-7p^{2}+8p+1}{p^{4}-4p^{3}+2p^{2}+4p-3}$$