Examples

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

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:

$2$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of

$2$ .

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Step 2.1.1

Cancel the common factor.

$f (x) = \frac{2 x^{2}}{2} + \frac{6}{2}$

Step 2.1.2

Divide

$x^{2}$ by

$1$ .

$f (x) = x^{2} + \frac{6}{2}$

Step 2.2

Divide

$6$ by

$2$ .

$f (x) = x^{2} + 3$

Step 3

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

$1$ .

$3$

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.

$b_{1} = | 3 |$

Step 4.2

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

$0$ and

$3$ is

$3$ .

$b_{1} = 3 + 1$

Step 4.3

Add

$3$ and

$1$ .

$b_{1} = 4$

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

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

$0$ and

$3$ is

$3$ .

$b_{2} = 3$

Step 5.2

Arrange the terms in ascending order.

$b_{2} = 1, 3$

Step 5.3

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

$b_{2} = 3$

Step 6

Take the smaller bound option between

$b_{1} = 4$ and

$b_{2} = 3$ .

Smaller Bound:

$3$

Step 7

Every real root on

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

$- 3$ and

$3$ .

$- 3$ and

$3$

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