Pre-Algebra Examples

Find the Bounds of the Zeros p(x)=6x^4+19x^3+24x^2+24x+8
Step 1
Check the leading coefficient of the function. This number is the coefficient of the expression with the largest degree.
Largest Degree:
Leading Coefficient:
Step 2
Simplify each term.
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Step 2.1
Cancel the common factor of .
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Step 2.1.1
Cancel the common factor.
Step 2.1.2
Divide by .
Step 2.2
Cancel the common factor of and .
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Step 2.2.1
Factor out of .
Step 2.2.2
Cancel the common factors.
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Step 2.2.2.1
Factor out of .
Step 2.2.2.2
Cancel the common factor.
Step 2.2.2.3
Rewrite the expression.
Step 2.2.2.4
Divide by .
Step 2.3
Cancel the common factor of and .
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Step 2.3.1
Factor out of .
Step 2.3.2
Cancel the common factors.
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Step 2.3.2.1
Factor out of .
Step 2.3.2.2
Cancel the common factor.
Step 2.3.2.3
Rewrite the expression.
Step 2.3.2.4
Divide by .
Step 2.4
Cancel the common factor of and .
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Step 2.4.1
Factor out of .
Step 2.4.2
Cancel the common factors.
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Step 2.4.2.1
Factor out of .
Step 2.4.2.2
Cancel the common factor.
Step 2.4.2.3
Rewrite the expression.
Step 3
Create a list of the coefficients of the function except the leading coefficient of .
Step 4
There will be two bound options, and , the smaller of which is the answer. To calculate the first bound option, find the absolute value of the largest coefficient from the list of coefficients. Then add .
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Step 4.1
Arrange the terms in ascending order.
Step 4.2
The maximum value is the largest value in the arranged data set.
Step 4.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.4
Add and .
Step 5
To calculate the second bound option, sum the absolute values of the coefficients from the list of coefficients. If the sum is greater than , use that number. If not, use .
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Step 5.1
Simplify each term.
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Step 5.1.1
is approximately which is positive so remove the absolute value
Step 5.1.2
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.1.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.1.4
is approximately which is positive so remove the absolute value
Step 5.2
Find the common denominator.
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Step 5.2.1
Write as a fraction with denominator .
Step 5.2.2
Multiply by .
Step 5.2.3
Multiply by .
Step 5.2.4
Write as a fraction with denominator .
Step 5.2.5
Multiply by .
Step 5.2.6
Multiply by .
Step 5.2.7
Multiply by .
Step 5.2.8
Multiply by .
Step 5.2.9
Reorder the factors of .
Step 5.2.10
Multiply by .
Step 5.3
Combine the numerators over the common denominator.
Step 5.4
Simplify each term.
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Step 5.4.1
Multiply by .
Step 5.4.2
Multiply by .
Step 5.4.3
Multiply by .
Step 5.5
Reduce the expression by cancelling the common factors.
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Step 5.5.1
Add and .
Step 5.5.2
Simplify by adding numbers.
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Step 5.5.2.1
Add and .
Step 5.5.2.2
Add and .
Step 5.5.3
Cancel the common factor of and .
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Step 5.5.3.1
Factor out of .
Step 5.5.3.2
Cancel the common factors.
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Step 5.5.3.2.1
Factor out of .
Step 5.5.3.2.2
Cancel the common factor.
Step 5.5.3.2.3
Rewrite the expression.
Step 5.6
Arrange the terms in ascending order.
Step 5.7
The maximum value is the largest value in the arranged data set.
Step 6
Take the smaller bound option between and .
Smaller Bound:
Step 7
Every real root on lies between and .
and