Pre-Algebra Examples

$f (x) = 6 - 8 x$

Step 1

Check the leading coefficient of the function. This number is the coefficient of the expression with the largest degree.

Largest Degree: $1$

Leading Coefficient: $- 8$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $6$ and $- 8$ .

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Step 2.1.1

Factor $2$ out of $6$ .

$f (x) = \frac{2 (3)}{- 8} + \frac{- 8 x}{- 8}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $2$ out of $- 8$ .

$f (x) = \frac{2 \cdot 3}{2 \cdot - 4} + \frac{- 8 x}{- 8}$

Step 2.1.2.2

Cancel the common factor.

$f (x) = \frac{2 \cdot 3}{2 \cdot - 4} + \frac{- 8 x}{- 8}$

Step 2.1.2.3

Rewrite the expression.

$f (x) = \frac{3}{- 4} + \frac{- 8 x}{- 8}$

Step 2.2

Move the negative in front of the fraction.

$f (x) = - \frac{3}{4} + \frac{- 8 x}{- 8}$

Step 2.3

Cancel the common factor of $- 8$ .

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Step 2.3.1

Cancel the common factor.

$f (x) = - \frac{3}{4} + \frac{- 8 x}{- 8}$

Step 2.3.2

Divide $x$ by $1$ .

$f (x) = - \frac{3}{4} + x$

Step 3

Create a list of the coefficients of the function except the leading coefficient of $1$ .

$- \frac{3}{4}$

Step 4

There will be two bound options, $b_{1}$ and $b_{2}$ , 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 $1$ .

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Step 4.1

Arrange the terms in ascending order.

$b_{1} = | - \frac{3}{4} |$

Step 4.2

$- \frac{3}{4}$ is approximately $- 0.75$ which is negative so negate $- \frac{3}{4}$ and remove the absolute value

$b_{1} = \frac{3}{4} + 1$

Step 4.3

Write $1$ as a fraction with a common denominator.

$b_{1} = \frac{3}{4} + \frac{4}{4}$

Step 4.4

Combine the numerators over the common denominator.

$b_{1} = \frac{3 + 4}{4}$

Step 4.5

Add $3$ and $4$ .

$b_{1} = \frac{7}{4}$

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 $1$ , use that number. If not, use $1$ .

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Step 5.1

$- \frac{3}{4}$ is approximately $- 0.75$ which is negative so negate $- \frac{3}{4}$ and remove the absolute value

$b_{2} = \frac{3}{4}$

Step 5.2

Arrange the terms in ascending order.

$b_{2} = \frac{3}{4}, 1$

Step 5.3

The maximum value is the largest value in the arranged data set.

$b_{2} = 1$

Step 6

Take the smaller bound option between $b_{1} = \frac{7}{4}$ and $b_{2} = 1$ .

Smaller Bound: $1$

Step 7

Every real root on $f (x) = 6 - 8 x$ lies between $- 1$ and $1$ .

$- 1$ and $1$