Pre-Algebra Examples

$x^{2} + 12 x$

Step 1

Write $x^{2} + 12 x$ as a function.

$f (x) = x^{2} + 12 x$

Step 2

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

Largest Degree: $2$

Leading Coefficient: $1$

Step 3

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

$12$

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.

$b_{1} = | 12 |$

Step 4.2

The absolute value is the distance between a number and zero. The distance between $0$ and $12$ is $12$ .

$b_{1} = 12 + 1$

Step 4.3

Add $12$ and $1$ .

$b_{1} = 13$

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

The absolute value is the distance between a number and zero. The distance between $0$ and $12$ is $12$ .

$b_{2} = 12$

Step 5.2

Arrange the terms in ascending order.

$b_{2} = 1, 12$

Step 5.3

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

$b_{2} = 12$

Step 6

Take the smaller bound option between $b_{1} = 13$ and $b_{2} = 12$ .

Smaller Bound: $12$

Step 7

Every real root on $f (x) = x^{2} + 12 x$ lies between $- 12$ and $12$ .

$- 12$ and $12$