Solve frac{2x^3-7x^2+7x-2/4x^4-13x^2+3/6{x^3-31{x}^2+3x+10}}{{x}^3-6{x}^2+3x+10}

$$\frac{ \frac{ \frac{ 2 { x }^{ 3 } -7 { x }^{ 2 } +7x-2 }{ 4 { x }^{ 4 } -13 { x }^{ 2 } +3 } }{ 6 { x }^{ 3 } -31 { x }^{ 2 } +3x+10 } }{ { x }^{ 3 } -6 { x }^{ 2 } +3x+10 }$$

Evaluate

$$\frac{x-1}{\left(3x-2\right)\left(x+1\right)\left(x^{2}-3\right)\left(\left(x-5\right)\left(2x+1\right)\right)^{2}}$$

Short Solution Steps

Express \frac{\frac{\frac{2x^{3}-7x^{2}+7x-2}{4x^{4}-13x^{2}+3}}{6x^{3}-31x^{2}+3x+10}}{x^{3}-6x^{2}+3x+10} as a single fraction.

$$\frac{\frac{2x^{3}-7x^{2}+7x-2}{4x^{4}-13x^{2}+3}}{\left(6x^{3}-31x^{2}+3x+10\right)\left(x^{3}-6x^{2}+3x+10\right)}$$

Factor the expressions that are not already factored in \frac{2x^{3}-7x^{2}+7x-2}{4x^{4}-13x^{2}+3}.

$$\frac{\frac{2\left(x-1\right)\left(\frac{1}{2}x-1\right)\left(2x-1\right)}{\left(2x-1\right)\left(2x+1\right)\left(x^{2}-3\right)}}{\left(6x^{3}-31x^{2}+3x+10\right)\left(x^{3}-6x^{2}+3x+10\right)}$$

Cancel out 2x-1 in both numerator and denominator.

$$\frac{\frac{2\left(x-1\right)\left(\frac{1}{2}x-1\right)}{\left(2x+1\right)\left(x^{2}-3\right)}}{\left(6x^{3}-31x^{2}+3x+10\right)\left(x^{3}-6x^{2}+3x+10\right)}$$

Express \frac{\frac{2\left(x-1\right)\left(\frac{1}{2}x-1\right)}{\left(2x+1\right)\left(x^{2}-3\right)}}{\left(6x^{3}-31x^{2}+3x+10\right)\left(x^{3}-6x^{2}+3x+10\right)} as a single fraction.

$$\frac{2\left(x-1\right)\left(\frac{1}{2}x-1\right)}{\left(2x+1\right)\left(x^{2}-3\right)\left(6x^{3}-31x^{2}+3x+10\right)\left(x^{3}-6x^{2}+3x+10\right)}$$

Factor the expressions that are not already factored.

$$\frac{\frac{1}{2}\times 2\left(x-2\right)\left(x-1\right)}{2\left(x-2\right)\left(\frac{3}{2}x-1\right)\left(x+1\right)\left(x-5\right)^{2}\left(x^{2}-3\right)\left(2x+1\right)^{2}}$$

Cancel out 2\left(x-2\right) in both numerator and denominator.

$$\frac{\frac{1}{2}\left(x-1\right)}{\left(\frac{3}{2}x-1\right)\left(x+1\right)\left(x-5\right)^{2}\left(x^{2}-3\right)\left(2x+1\right)^{2}}$$

Expand the expression.

$$\frac{\frac{1}{2}x-\frac{1}{2}}{6x^{8}-52x^{7}+\frac{103}{2}x^{6}+\frac{715}{2}x^{5}-187x^{4}-682x^{3}-\frac{179}{2}x^{2}+\frac{465}{2}x+75}$$

Factor the expressions that are not already factored.

$$\frac{\frac{1}{2}\left(x-1\right)}{\frac{1}{2}\times 2\left(3x-2\right)\left(\frac{1}{2}x+\frac{1}{2}\right)\left(x-5\right)^{2}\left(x^{2}-3\right)\left(2x+1\right)^{2}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{x-1}{2\times \left(\frac{1}{2}\right)^{0}\left(3x-2\right)\left(\frac{1}{2}x+\frac{1}{2}\right)\left(x-5\right)^{2}\left(x^{2}-3\right)\left(2x+1\right)^{2}}$$

Expand the expression.

$$\frac{x-1}{12x^{8}-104x^{7}+103x^{6}+715x^{5}-374x^{4}-1364x^{3}-179x^{2}+465x+150}$$

Expand

$$\frac{x-1}{\left(x^{2}-4x-5\right)\left(2x^{3}+x^{2}-6x-3\right)\left(6x^{3}-31x^{2}+3x+10\right)}$$

Short Solution Steps

Express \frac{\frac{\frac{2x^{3}-7x^{2}+7x-2}{4x^{4}-13x^{2}+3}}{6x^{3}-31x^{2}+3x+10}}{x^{3}-6x^{2}+3x+10} as a single fraction.

$$\frac{\frac{2x^{3}-7x^{2}+7x-2}{4x^{4}-13x^{2}+3}}{\left(6x^{3}-31x^{2}+3x+10\right)\left(x^{3}-6x^{2}+3x+10\right)}$$

Factor the expressions that are not already factored in \frac{2x^{3}-7x^{2}+7x-2}{4x^{4}-13x^{2}+3}.

$$\frac{\frac{2\left(x-1\right)\left(\frac{1}{2}x-1\right)\left(2x-1\right)}{\left(2x-1\right)\left(2x+1\right)\left(x^{2}-3\right)}}{\left(6x^{3}-31x^{2}+3x+10\right)\left(x^{3}-6x^{2}+3x+10\right)}$$

Cancel out 2x-1 in both numerator and denominator.

$$\frac{\frac{2\left(x-1\right)\left(\frac{1}{2}x-1\right)}{\left(2x+1\right)\left(x^{2}-3\right)}}{\left(6x^{3}-31x^{2}+3x+10\right)\left(x^{3}-6x^{2}+3x+10\right)}$$

Express \frac{\frac{2\left(x-1\right)\left(\frac{1}{2}x-1\right)}{\left(2x+1\right)\left(x^{2}-3\right)}}{\left(6x^{3}-31x^{2}+3x+10\right)\left(x^{3}-6x^{2}+3x+10\right)} as a single fraction.

$$\frac{2\left(x-1\right)\left(\frac{1}{2}x-1\right)}{\left(2x+1\right)\left(x^{2}-3\right)\left(6x^{3}-31x^{2}+3x+10\right)\left(x^{3}-6x^{2}+3x+10\right)}$$

Factor the expressions that are not already factored.

$$\frac{\frac{1}{2}\times 2\left(x-2\right)\left(x-1\right)}{2\left(x-2\right)\left(\frac{3}{2}x-1\right)\left(x+1\right)\left(x-5\right)^{2}\left(x^{2}-3\right)\left(2x+1\right)^{2}}$$

Cancel out 2\left(x-2\right) in both numerator and denominator.

$$\frac{\frac{1}{2}\left(x-1\right)}{\left(\frac{3}{2}x-1\right)\left(x+1\right)\left(x-5\right)^{2}\left(x^{2}-3\right)\left(2x+1\right)^{2}}$$

Expand the expression.

$$\frac{\frac{1}{2}x-\frac{1}{2}}{6x^{8}-52x^{7}+\frac{103}{2}x^{6}+\frac{715}{2}x^{5}-187x^{4}-682x^{3}-\frac{179}{2}x^{2}+\frac{465}{2}x+75}$$

Factor the expressions that are not already factored.

$$\frac{\frac{1}{2}\left(x-1\right)}{\frac{1}{2}\times 2\left(3x-2\right)\left(\frac{1}{2}x+\frac{1}{2}\right)\left(x-5\right)^{2}\left(x^{2}-3\right)\left(2x+1\right)^{2}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{x-1}{2\times \left(\frac{1}{2}\right)^{0}\left(3x-2\right)\left(\frac{1}{2}x+\frac{1}{2}\right)\left(x-5\right)^{2}\left(x^{2}-3\right)\left(2x+1\right)^{2}}$$

Expand the expression.

$$\frac{x-1}{12x^{8}-104x^{7}+103x^{6}+715x^{5}-374x^{4}-1364x^{3}-179x^{2}+465x+150}$$