Algebra Examples

Prove that a Root is on the Interval
,
Step 1
The Intermediate Value Theorem states that, if is a real-valued continuous function on the interval , and is a number between and , then there is a contained in the interval such that .
Step 2
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation:
Set-Builder Notation:
Step 3
Calculate .
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Step 3.1
Remove parentheses.
Step 3.2
Simplify each term.
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Step 3.2.1
Raise to the power of .
Step 3.2.2
Multiply by .
Step 3.3
Subtract from .
Step 4
Calculate .
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Step 4.1
Remove parentheses.
Step 4.2
Simplify each term.
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Step 4.2.1
Raise to the power of .
Step 4.2.2
Multiply by .
Step 4.3
Add and .
Step 5
is not on the interval .
There is no root on the interval.
Step 6
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