Precalculus Examples

Find the Roots (Zeros) f(x)=x^5-5x^3+4x
Step 1
Set equal to .
Step 2
Solve for .
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Step 2.1
Factor the left side of the equation.
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Step 2.1.1
Factor out of .
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Step 2.1.1.1
Factor out of .
Step 2.1.1.2
Factor out of .
Step 2.1.1.3
Factor out of .
Step 2.1.1.4
Factor out of .
Step 2.1.1.5
Factor out of .
Step 2.1.2
Rewrite as .
Step 2.1.3
Let . Substitute for all occurrences of .
Step 2.1.4
Factor using the AC method.
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Step 2.1.4.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 2.1.4.2
Write the factored form using these integers.
Step 2.1.5
Replace all occurrences of with .
Step 2.1.6
Rewrite as .
Step 2.1.7
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2.1.8
Rewrite as .
Step 2.1.9
Factor.
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Step 2.1.9.1
Factor.
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Step 2.1.9.1.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2.1.9.1.2
Remove unnecessary parentheses.
Step 2.1.9.2
Remove unnecessary parentheses.
Step 2.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.3
Set equal to .
Step 2.4
Set equal to and solve for .
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Step 2.4.1
Set equal to .
Step 2.4.2
Subtract from both sides of the equation.
Step 2.5
Set equal to and solve for .
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Step 2.5.1
Set equal to .
Step 2.5.2
Add to both sides of the equation.
Step 2.6
Set equal to and solve for .
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Step 2.6.1
Set equal to .
Step 2.6.2
Subtract from both sides of the equation.
Step 2.7
Set equal to and solve for .
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Step 2.7.1
Set equal to .
Step 2.7.2
Add to both sides of the equation.
Step 2.8
The final solution is all the values that make true.
Step 3