Pre-Algebra Examples

Find the Bounds of the Zeros f(x)=4x(x-7)(x+12)
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
The leading coefficient needs to be . If it is not, divide the expression by it to make it .
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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
Apply the distributive property.
Step 2.3
Multiply by .
Step 2.4
Move to the left of .
Step 2.5
Expand using the FOIL Method.
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Step 2.5.1
Apply the distributive property.
Step 2.5.2
Apply the distributive property.
Step 2.5.3
Apply the distributive property.
Step 2.6
Simplify and combine like terms.
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Step 2.6.1
Simplify each term.
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Step 2.6.1.1
Multiply by by adding the exponents.
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Step 2.6.1.1.1
Multiply by .
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Step 2.6.1.1.1.1
Raise to the power of .
Step 2.6.1.1.1.2
Use the power rule to combine exponents.
Step 2.6.1.1.2
Add and .
Step 2.6.1.2
Move to the left of .
Step 2.6.1.3
Multiply by by adding the exponents.
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Step 2.6.1.3.1
Move .
Step 2.6.1.3.2
Multiply by .
Step 2.6.1.4
Multiply by .
Step 2.6.2
Subtract from .
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
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.1.2
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.2
Add and .
Step 5.3
Arrange the terms in ascending order.
Step 5.4
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