Pre-Algebra Examples

$f (x) = \frac{25000 (x - 14)}{x^{2} - 9}$

Step 1

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

Largest Degree: $- 1$

Leading Coefficient: $25000$

Step 2

The leading coefficient needs to be $1$ . If it is not, divide the expression by it to make it $1$ .

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$f (x) = \frac{25000 (x - 14)}{x^{2} - 9} \cdot \frac{1}{25000}$

Step 2.2

Cancel the common factor of $25000$ .

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Step 2.2.1

Cancel the common factor.

$f (x) = \frac{25000 (x - 14)}{x^{2} - 9} \cdot \frac{1}{25000}$

Step 2.2.2

Rewrite the expression.

$f (x) = \frac{x - 14}{x^{2} - 9}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Rewrite $9$ as $3^{2}$ .

$f (x) = \frac{x - 14}{x^{2} - 3^{2}}$

Step 2.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$f (x) = \frac{x - 14}{(x + 3) (x - 3)}$

Step 3

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

$0$

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.

$b_{1} = 0$

Step 4.2

Add $0$ and $1$ .

$b_{1} = 1$

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

Arrange the terms in ascending order.

$b_{2} = 0, 1$

Step 5.2

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

$b_{2} = 1$

Step 6

The bound options are the same.

Bound: $1$

Step 7

Every real root on $f (x) = \frac{25000 (x - 14)}{x^{2} - 9}$ lies between $- 1$ and $1$ .

$- 1$ and $1$