Trigonometry Examples

Split Using Partial Fraction Decomposition (x^-6)/(x^3)
Step 1
Decompose the fraction and multiply through by the common denominator.
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Step 1.1
Reduce the expression by cancelling the common factors.
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Step 1.1.1
Multiply by .
Step 1.1.2
Factor out of .
Step 1.1.3
Cancel the common factor.
Step 1.1.4
Rewrite the expression.
Step 1.2
Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .
Step 1.3
Cancel the common factor of .
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Step 1.3.1
Cancel the common factor.
Step 1.3.2
Rewrite the expression.
Step 1.4
Simplify each term.
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Step 1.4.1
Cancel the common factor of .
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Step 1.4.1.1
Cancel the common factor.
Step 1.4.1.2
Divide by .
Step 1.4.2
Cancel the common factor of and .
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Step 1.4.2.1
Factor out of .
Step 1.4.2.2
Cancel the common factors.
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Step 1.4.2.2.1
Multiply by .
Step 1.4.2.2.2
Cancel the common factor.
Step 1.4.2.2.3
Rewrite the expression.
Step 1.4.2.2.4
Divide by .
Step 1.4.3
Cancel the common factor of and .
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Step 1.4.3.1
Factor out of .
Step 1.4.3.2
Cancel the common factors.
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Step 1.4.3.2.1
Multiply by .
Step 1.4.3.2.2
Cancel the common factor.
Step 1.4.3.2.3
Rewrite the expression.
Step 1.4.3.2.4
Divide by .
Step 1.4.4
Cancel the common factor of and .
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Step 1.4.4.1
Factor out of .
Step 1.4.4.2
Cancel the common factors.
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Step 1.4.4.2.1
Multiply by .
Step 1.4.4.2.2
Cancel the common factor.
Step 1.4.4.2.3
Rewrite the expression.
Step 1.4.4.2.4
Divide by .
Step 1.4.5
Cancel the common factor of and .
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Step 1.4.5.1
Factor out of .
Step 1.4.5.2
Cancel the common factors.
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Step 1.4.5.2.1
Multiply by .
Step 1.4.5.2.2
Cancel the common factor.
Step 1.4.5.2.3
Rewrite the expression.
Step 1.4.5.2.4
Divide by .
Step 1.4.6
Cancel the common factor of and .
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Step 1.4.6.1
Factor out of .
Step 1.4.6.2
Cancel the common factors.
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Step 1.4.6.2.1
Multiply by .
Step 1.4.6.2.2
Cancel the common factor.
Step 1.4.6.2.3
Rewrite the expression.
Step 1.4.6.2.4
Divide by .
Step 1.4.7
Cancel the common factor of and .
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Step 1.4.7.1
Factor out of .
Step 1.4.7.2
Cancel the common factors.
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Step 1.4.7.2.1
Multiply by .
Step 1.4.7.2.2
Cancel the common factor.
Step 1.4.7.2.3
Rewrite the expression.
Step 1.4.7.2.4
Divide by .
Step 1.4.8
Cancel the common factor of and .
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Step 1.4.8.1
Factor out of .
Step 1.4.8.2
Cancel the common factors.
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Step 1.4.8.2.1
Multiply by .
Step 1.4.8.2.2
Cancel the common factor.
Step 1.4.8.2.3
Rewrite the expression.
Step 1.4.8.2.4
Divide by .
Step 1.4.9
Cancel the common factor of and .
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Step 1.4.9.1
Factor out of .
Step 1.4.9.2
Cancel the common factors.
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Step 1.4.9.2.1
Raise to the power of .
Step 1.4.9.2.2
Factor out of .
Step 1.4.9.2.3
Cancel the common factor.
Step 1.4.9.2.4
Rewrite the expression.
Step 1.4.9.2.5
Divide by .
Step 1.5
Simplify the expression.
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Step 1.5.1
Reorder and .
Step 1.5.2
Reorder and .
Step 1.5.3
Reorder and .
Step 1.5.4
Reorder and .
Step 1.5.5
Move .
Step 1.5.6
Move .
Step 1.5.7
Move .
Step 1.5.8
Move .
Step 1.5.9
Move .
Step 1.5.10
Move .
Step 1.5.11
Move .
Step 1.5.12
Reorder and .
Step 2
Create equations for the partial fraction variables and use them to set up a system of equations.
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Step 2.1
Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 2.2
Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 2.3
Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 2.4
Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 2.5
Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 2.6
Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 2.7
Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 2.8
Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 2.9
Create an equation for the partial fraction variables by equating the coefficients of the terms not containing . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 2.10
Set up the system of equations to find the coefficients of the partial fractions.
Step 3
Solve the system of equations.
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Step 3.1
Rewrite the equation as .
Step 3.2
Replace all occurrences of with in each equation.
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Step 3.2.1
Rewrite the equation as .
Step 3.2.2
Rewrite the equation as .
Step 3.2.3
Rewrite the equation as .
Step 3.2.4
Rewrite the equation as .
Step 3.2.5
Rewrite the equation as .
Step 3.2.6
Rewrite the equation as .
Step 3.2.7
Rewrite the equation as .
Step 3.2.8
Rewrite the equation as .
Step 3.3
List all of the solutions.
Step 4
Replace each of the partial fraction coefficients in with the values found for , , , , , , , , and .