Finite Math Examples

$f (x) = x^{2} - 5 x + 6$

Step 1

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 2

Create a list of the coefficients of the function except the leading coefficient of

$1$ .

$- 5, 6$

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.

$b_{1} = | - 5 |, | 6 |$

Step 3.2

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

$b_{1} = | 6 |$

Step 3.3

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

$0$ and

$6$ is

$6$ .

$b_{1} = 6 + 1$

Step 3.4

Add

$6$ and

$1$ .

$b_{1} = 7$

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

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

$- 5$ and

$0$ is

$5$ .

$b_{2} = 5 + | 6 |$

Step 4.1.2

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

$0$ and

$6$ is

$6$ .

$b_{2} = 5 + 6$

Step 4.2

Add

$5$ and

$6$ .

$b_{2} = 11$

Step 4.3

Arrange the terms in ascending order.

$b_{2} = 1, 11$

Step 4.4

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

$b_{2} = 11$

Step 5

Take the smaller bound option between

$b_{1} = 7$ and

$b_{2} = 11$ .

Smaller Bound:

$7$

Step 6

Every real root on

$f (x) = x^{2} - 5 x + 6$ lies between

$- 7$ and

$7$ .

$- 7$ and

$7$

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