Solve -3x/x^{3-4x^{2}+5x-2}+x+7/x^{2-5x+6}-frac{x^{3}+x^2-4x+23}{x^4-7x^3+17x^2-17x+6}

$$\frac { - 3 x } { x ^ { 3 } - 4 x ^ { 2 } + 5 x - 2 } + \frac { x + 7 } { x ^ { 2 } - 5 x + 6 } - \frac { x ^ { 3 } + x ^ { 2 } - 4 x + 23 } { x ^ { 4 } - 7 x ^ { 3 } + 17 x ^ { 2 } - 17 x + 6 }$$

Evaluate

$$\frac{x^{2}-16}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}$$

Short Solution Steps

Factor x^{3}-4x^{2}+5x-2. Factor x^{2}-5x+6.

$$\frac{-3x}{\left(x-2\right)\left(x-1\right)^{2}}+\frac{x+7}{\left(x-3\right)\left(x-2\right)}-\frac{x^{3}+x^{2}-4x+23}{x^{4}-7x^{3}+17x^{2}-17x+6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x-1\right)^{2} and \left(x-3\right)\left(x-2\right) is \left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}. Multiply \frac{-3x}{\left(x-2\right)\left(x-1\right)^{2}} times \frac{x-3}{x-3}. Multiply \frac{x+7}{\left(x-3\right)\left(x-2\right)} times \frac{\left(x-1\right)^{2}}{\left(x-1\right)^{2}}.

$$\frac{-3x\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}+\frac{\left(x+7\right)\left(x-1\right)^{2}}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}-\frac{x^{3}+x^{2}-4x+23}{x^{4}-7x^{3}+17x^{2}-17x+6}$$

Since \frac{-3x\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}} and \frac{\left(x+7\right)\left(x-1\right)^{2}}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-3x\left(x-3\right)+\left(x+7\right)\left(x-1\right)^{2}}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}-\frac{x^{3}+x^{2}-4x+23}{x^{4}-7x^{3}+17x^{2}-17x+6}$$

Do the multiplications in -3x\left(x-3\right)+\left(x+7\right)\left(x-1\right)^{2}.

$$\frac{-3x^{2}+9x+x^{3}-2x^{2}+x+7x^{2}-14x+7}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}-\frac{x^{3}+x^{2}-4x+23}{x^{4}-7x^{3}+17x^{2}-17x+6}$$

Combine like terms in -3x^{2}+9x+x^{3}-2x^{2}+x+7x^{2}-14x+7.

$$\frac{2x^{2}-4x+x^{3}+7}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}-\frac{x^{3}+x^{2}-4x+23}{x^{4}-7x^{3}+17x^{2}-17x+6}$$

Factor x^{4}-7x^{3}+17x^{2}-17x+6.

$$\frac{2x^{2}-4x+x^{3}+7}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}-\frac{x^{3}+x^{2}-4x+23}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}$$

Since \frac{2x^{2}-4x+x^{3}+7}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}} and \frac{x^{3}+x^{2}-4x+23}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x^{2}-4x+x^{3}+7-\left(x^{3}+x^{2}-4x+23\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}$$

Do the multiplications in 2x^{2}-4x+x^{3}+7-\left(x^{3}+x^{2}-4x+23\right).

$$\frac{2x^{2}-4x+x^{3}+7-x^{3}-x^{2}+4x-23}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}$$

Combine like terms in 2x^{2}-4x+x^{3}+7-x^{3}-x^{2}+4x-23.

$$\frac{x^{2}-16}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}$$

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

$$\frac{x^{2}-16}{x^{4}-7x^{3}+17x^{2}-17x+6}$$

Expand

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

Short Solution Steps

Factor x^{3}-4x^{2}+5x-2. Factor x^{2}-5x+6.

$$\frac{-3x}{\left(x-2\right)\left(x-1\right)^{2}}+\frac{x+7}{\left(x-3\right)\left(x-2\right)}-\frac{x^{3}+x^{2}-4x+23}{x^{4}-7x^{3}+17x^{2}-17x+6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x-1\right)^{2} and \left(x-3\right)\left(x-2\right) is \left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}. Multiply \frac{-3x}{\left(x-2\right)\left(x-1\right)^{2}} times \frac{x-3}{x-3}. Multiply \frac{x+7}{\left(x-3\right)\left(x-2\right)} times \frac{\left(x-1\right)^{2}}{\left(x-1\right)^{2}}.

$$\frac{-3x\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}+\frac{\left(x+7\right)\left(x-1\right)^{2}}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}-\frac{x^{3}+x^{2}-4x+23}{x^{4}-7x^{3}+17x^{2}-17x+6}$$

Since \frac{-3x\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}} and \frac{\left(x+7\right)\left(x-1\right)^{2}}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-3x\left(x-3\right)+\left(x+7\right)\left(x-1\right)^{2}}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}-\frac{x^{3}+x^{2}-4x+23}{x^{4}-7x^{3}+17x^{2}-17x+6}$$

Do the multiplications in -3x\left(x-3\right)+\left(x+7\right)\left(x-1\right)^{2}.

$$\frac{-3x^{2}+9x+x^{3}-2x^{2}+x+7x^{2}-14x+7}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}-\frac{x^{3}+x^{2}-4x+23}{x^{4}-7x^{3}+17x^{2}-17x+6}$$

Combine like terms in -3x^{2}+9x+x^{3}-2x^{2}+x+7x^{2}-14x+7.

$$\frac{2x^{2}-4x+x^{3}+7}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}-\frac{x^{3}+x^{2}-4x+23}{x^{4}-7x^{3}+17x^{2}-17x+6}$$

Factor x^{4}-7x^{3}+17x^{2}-17x+6.

$$\frac{2x^{2}-4x+x^{3}+7}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}-\frac{x^{3}+x^{2}-4x+23}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}$$

Since \frac{2x^{2}-4x+x^{3}+7}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}} and \frac{x^{3}+x^{2}-4x+23}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x^{2}-4x+x^{3}+7-\left(x^{3}+x^{2}-4x+23\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}$$

Do the multiplications in 2x^{2}-4x+x^{3}+7-\left(x^{3}+x^{2}-4x+23\right).

$$\frac{2x^{2}-4x+x^{3}+7-x^{3}-x^{2}+4x-23}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}$$

Combine like terms in 2x^{2}-4x+x^{3}+7-x^{3}-x^{2}+4x-23.

$$\frac{x^{2}-16}{\left(x-3\right)\left(x-2\right)\left(x-1\right)^{2}}$$

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

$$\frac{x^{2}-16}{x^{4}-7x^{3}+17x^{2}-17x+6}$$