Pre-Algebra Examples

$x^{2} - 15 x + 56$

Step 1

Write $x^{2} - 15 x + 56$ as a function.

$f (x) = x^{2} - 15 x + 56$

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 $1$ .

$- 15, 56$

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.

$b_{1} = | - 15 |, | 56 |$

Step 4.2

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

$b_{1} = | 56 |$

Step 4.3

The absolute value is the distance between a number and zero. The distance between $0$ and $56$ is $56$ .

$b_{1} = 56 + 1$

Step 4.4

Add $56$ and $1$ .

$b_{1} = 57$

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

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 $- 15$ and $0$ is $15$ .

$b_{2} = 15 + | 56 |$

Step 5.1.2

The absolute value is the distance between a number and zero. The distance between $0$ and $56$ is $56$ .

$b_{2} = 15 + 56$

Step 5.2

Add $15$ and $56$ .

$b_{2} = 71$

Step 5.3

Arrange the terms in ascending order.

$b_{2} = 1, 71$

Step 5.4

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

$b_{2} = 71$

Step 6

Take the smaller bound option between $b_{1} = 57$ and $b_{2} = 71$ .

Smaller Bound: $57$

Step 7

Every real root on $f (x) = x^{2} - 15 x + 56$ lies between $- 57$ and $57$ .

$- 57$ and $57$