Linear Algebra Examples

Find the Domain x^8+80x^4-81=0
Step 1
Factor the left side of the equation.
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Step 1.1
Rewrite as .
Step 1.2
Let . Substitute for all occurrences of .
Step 1.3
Factor using the AC method.
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Step 1.3.1
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 1.3.2
Write the factored form using these integers.
Step 1.4
Replace all occurrences of with .
Step 2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3
Set equal to and solve for .
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Step 3.1
Set equal to .
Step 3.2
Solve for .
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Step 3.2.1
Add to both sides of the equation.
Step 3.2.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.2.3
Any root of is .
Step 3.2.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 3.2.4.1
First, use the positive value of the to find the first solution.
Step 3.2.4.2
Next, use the negative value of the to find the second solution.
Step 3.2.4.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 4
Set equal to and solve for .
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Step 4.1
Set equal to .
Step 4.2
Solve for .
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Step 4.2.1
Subtract from both sides of the equation.
Step 4.2.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 4.2.3
Simplify .
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Step 4.2.3.1
Rewrite as .
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Step 4.2.3.1.1
Factor out of .
Step 4.2.3.1.2
Rewrite as .
Step 4.2.3.2
Pull terms out from under the radical.
Step 4.2.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 4.2.4.1
First, use the positive value of the to find the first solution.
Step 4.2.4.2
Next, use the negative value of the to find the second solution.
Step 4.2.4.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 5
The final solution is all the values that make true.
Step 6
The domain is the set of all valid values.
Step 7