Linear Algebra Examples

Find the Domain (5x+38x+21)/(3x^2+22x+7)
Step 1
Set the denominator in equal to to find where the expression is undefined.
Step 2
Solve for .
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Step 2.1
Factor by grouping.
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Step 2.1.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 2.1.1.1
Factor out of .
Step 2.1.1.2
Rewrite as plus
Step 2.1.1.3
Apply the distributive property.
Step 2.1.1.4
Multiply by .
Step 2.1.2
Factor out the greatest common factor from each group.
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Step 2.1.2.1
Group the first two terms and the last two terms.
Step 2.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 2.1.3
Factor the polynomial by factoring out the greatest common factor, .
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 and solve for .
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Step 2.3.1
Set equal to .
Step 2.3.2
Solve for .
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Step 2.3.2.1
Subtract from both sides of the equation.
Step 2.3.2.2
Divide each term in by and simplify.
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Step 2.3.2.2.1
Divide each term in by .
Step 2.3.2.2.2
Simplify the left side.
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Step 2.3.2.2.2.1
Cancel the common factor of .
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Step 2.3.2.2.2.1.1
Cancel the common factor.
Step 2.3.2.2.2.1.2
Divide by .
Step 2.3.2.2.3
Simplify the right side.
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Step 2.3.2.2.3.1
Move the negative in front of the fraction.
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
The final solution is all the values that make true.
Step 3
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Step 4