Pre-Algebra Examples

$f (x) = - 16 x^{2} + 42 (9) + 12$

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: $- 16$

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

Combine the numerators over the common denominator.

$f (x) = \frac{- 16 x^{2} + 42 (9) + 12}{- 16}$

Step 2.2

Multiply $42$ by $9$ .

$f (x) = \frac{- 16 x^{2} + 378 + 12}{- 16}$

Step 2.3

Add $378$ and $12$ .

$f (x) = \frac{- 16 x^{2} + 390}{- 16}$

Step 2.4

Cancel the common factor of $- 16 x^{2} + 390$ and $- 16$ .

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Step 2.4.1

Factor $2$ out of $- 16 x^{2}$ .

$f (x) = \frac{2 (- 8 x^{2}) + 390}{- 16}$

Step 2.4.2

Factor $2$ out of $390$ .

$f (x) = \frac{2 (- 8 x^{2}) + 2 (195)}{- 16}$

Step 2.4.3

Factor $2$ out of $2 (- 8 x^{2}) + 2 (195)$ .

$f (x) = \frac{2 (- 8 x^{2} + 195)}{- 16}$

Step 2.4.4

Cancel the common factors.

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Step 2.4.4.1

Factor $2$ out of $- 16$ .

$f (x) = \frac{2 (- 8 x^{2} + 195)}{2 (- 8)}$

Step 2.4.4.2

Cancel the common factor.

$f (x) = \frac{2 (- 8 x^{2} + 195)}{2 \cdot - 8}$

Step 2.4.4.3

Rewrite the expression.

$f (x) = \frac{- 8 x^{2} + 195}{- 8}$

Step 2.5

Move the negative in front of the fraction.

$f (x) = - \frac{- 8 x^{2} + 195}{8}$

Step 2.6

Factor $- 1$ out of $- 8 x^{2}$ .

$f (x) = - \frac{- (8 x^{2}) + 195}{8}$

Step 2.7

Rewrite $195$ as $- 1 (- 195)$ .

$f (x) = - \frac{- (8 x^{2}) - 1 \cdot - 195}{8}$

Step 2.8

Factor $- 1$ out of $- (8 x^{2}) - 1 (- 195)$ .

$f (x) = - \frac{- (8 x^{2} - 195)}{8}$

Step 2.9

Rewrite $- (8 x^{2} - 195)$ as $- 1 (8 x^{2} - 195)$ .

$f (x) = - \frac{- 1 (8 x^{2} - 195)}{8}$

Step 2.10

Move the negative in front of the fraction.

$f (x) = \frac{8 x^{2} - 195}{8}$

Step 2.11

Multiply $- 1$ by $- 1$ .

$f (x) = 1 (\frac{8 x^{2} - 195}{8})$

Step 2.12

Multiply $\frac{8 x^{2} - 195}{8}$ by $1$ .

$f (x) = \frac{8 x^{2} - 195}{8}$

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 $f (x) = - 16 x^{2} + 42 (9) + 12$ lies between $- 1$ and $1$ .

$- 1$ and $1$