Pre-Algebra Examples

$p (x) = - 1.2 x^{2} + 62.5 x - 491$

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: $- 1.2$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $- 1.2$ .

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Step 2.1.1

Cancel the common factor.

$p (x) = \frac{- 1.2 x^{2}}{- 1.2} + \frac{62.5 x}{- 1.2} + \frac{- 491}{- 1.2}$

Step 2.1.2

Divide $x^{2}$ by $1$ .

$p (x) = x^{2} + \frac{62.5 x}{- 1.2} + \frac{- 491}{- 1.2}$

Step 2.2

Move the negative in front of the fraction.

$p (x) = x^{2} - \frac{62.5 x}{1.2} + \frac{- 491}{- 1.2}$

Step 2.3

Factor $62.5$ out of $62.5 x$ .

$p (x) = x^{2} - \frac{62.5 (x)}{1.2} + \frac{- 491}{- 1.2}$

Step 2.4

Factor $1.2$ out of $1.2$ .

$p (x) = x^{2} - \frac{62.5 (x)}{1.2 (1)} + \frac{- 491}{- 1.2}$

Step 2.5

Separate fractions.

$p (x) = x^{2} - (\frac{62.5}{1.2} \cdot \frac{x}{1}) + \frac{- 491}{- 1.2}$

Step 2.6

Divide $62.5$ by $1.2$ .

$p (x) = x^{2} - (52.08\overline{3} (\frac{x}{1})) + \frac{- 491}{- 1.2}$

Step 2.7

Divide $x$ by $1$ .

$p (x) = x^{2} - (52.08\overline{3} x) + \frac{- 491}{- 1.2}$

Step 2.8

Multiply $52.08\overline{3}$ by $- 1$ .

$p (x) = x^{2} - 52.08\overline{3} x + \frac{- 491}{- 1.2}$

Step 2.9

Divide $- 491$ by $- 1.2$ .

$p (x) = x^{2} - 52.08\overline{3} x + 409.1\overline{6}$

Step 3

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

$- 52.08\overline{3}, 409.1\overline{6}$

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.

$b_{1} = | - 52.08\overline{3} |, | 409.1\overline{6} |$

Step 4.2

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

$b_{1} = | 409.1\overline{6} |$

Step 4.3

The absolute value is the distance between a number and zero. The distance between $0$ and $409.1\overline{6}$ is $409.1\overline{6}$ .

$b_{1} = 409.1\overline{6} + 1$

Step 4.4

Add $409.1\overline{6}$ and $1$ .

$b_{1} = 410.1\overline{6}$

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 $- 52.08\overline{3}$ and $0$ is $52.08\overline{3}$ .

$b_{2} = 52.08\overline{3} + | 409.1\overline{6} |$

Step 5.1.2

The absolute value is the distance between a number and zero. The distance between $0$ and $409.1\overline{6}$ is $409.1\overline{6}$ .

$b_{2} = 52.08\overline{3} + 409.1\overline{6}$

Step 5.2

Add $52.08\overline{3}$ and $409.1\overline{6}$ .

$b_{2} = 461.25$

Step 5.3

Arrange the terms in ascending order.

$b_{2} = 1, 461.25$

Step 5.4

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

$b_{2} = 461.25$

Step 6

Take the smaller bound option between $b_{1} = 410.1\overline{6}$ and $b_{2} = 461.25$ .

Smaller Bound: $410.1\overline{6}$

Step 7

Every real root on $p (x) = - 1.2 x^{2} + 62.5 x - 491$ lies between $- 410.1\overline{6}$ and $410.1\overline{6}$ .

$- 410.1\overline{6}$ and $410.1\overline{6}$