Pre-Algebra Examples

$g (x) = \frac{1}{4} x^{3} + 3$

Step 1

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

Largest Degree: $3$

Leading Coefficient: $1$

Step 2

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

$\frac{1}{4}, 3$

Step 3

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 3.1

Arrange the terms in ascending order.

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

Step 3.2

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

$b_{1} = | 3 |$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$b_{1} = 3 + 1$

Step 3.4

Add $3$ and $1$ .

$b_{1} = 4$

Step 4

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 4.1

Simplify each term.

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Step 4.1.1

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$b_{2} = \frac{1}{4} + | 3 |$

Step 4.1.2

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

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

Step 4.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$b_{2} = \frac{1}{4} + 3 \cdot \frac{4}{4}$

Step 4.3

Combine $3$ and $\frac{4}{4}$ .

$b_{2} = \frac{1}{4} + \frac{3 \cdot 4}{4}$

Step 4.4

Combine the numerators over the common denominator.

$b_{2} = \frac{1 + 3 \cdot 4}{4}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $3$ by $4$ .

$b_{2} = \frac{1 + 12}{4}$

Step 4.5.2

Add $1$ and $12$ .

$b_{2} = \frac{13}{4}$

Step 4.6

Arrange the terms in ascending order.

$b_{2} = 1, \frac{13}{4}$

Step 4.7

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

$b_{2} = \frac{13}{4}$

Step 5

Take the smaller bound option between $b_{1} = 4$ and $b_{2} = \frac{13}{4}$ .

Smaller Bound: $\frac{13}{4}$

Step 6

Every real root on $g (x) = \frac{1}{4} x^{3} + 3$ lies between $- \frac{13}{4}$ and $\frac{13}{4}$ .

$- \frac{13}{4}$ and $\frac{13}{4}$