Pre-Algebra Examples

$f (x) = - 2 x^{4} + 25 x^{3} - 95 x^{2} + 90 x + 72$

Step 1

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

Largest Degree: $4$

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.

$f (x) = \frac{- 2 x^{4}}{- 2} + \frac{25 x^{3}}{- 2} + \frac{- 95 x^{2}}{- 2} + \frac{90 x}{- 2} + \frac{72}{- 2}$

Step 2.1.2

Divide $x^{4}$ by $1$ .

$f (x) = x^{4} + \frac{25 x^{3}}{- 2} + \frac{- 95 x^{2}}{- 2} + \frac{90 x}{- 2} + \frac{72}{- 2}$

Step 2.2

Move the negative in front of the fraction.

$f (x) = x^{4} - \frac{25 x^{3}}{2} + \frac{- 95 x^{2}}{- 2} + \frac{90 x}{- 2} + \frac{72}{- 2}$

Step 2.3

Dividing two negative values results in a positive value.

$f (x) = x^{4} - \frac{25 x^{3}}{2} + \frac{95 x^{2}}{2} + \frac{90 x}{- 2} + \frac{72}{- 2}$

Step 2.4

Cancel the common factor of $90$ and $- 2$ .

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Step 2.4.1

Factor $2$ out of $90 x$ .

$f (x) = x^{4} - \frac{25 x^{3}}{2} + \frac{95 x^{2}}{2} + \frac{2 (45 x)}{- 2} + \frac{72}{- 2}$

Step 2.4.2

Move the negative one from the denominator of $\frac{45 x}{- 1}$ .

$f (x) = x^{4} - \frac{25 x^{3}}{2} + \frac{95 x^{2}}{2} - 1 \cdot (45 x) + \frac{72}{- 2}$

Step 2.5

Rewrite $- 1 \cdot (45 x)$ as $- (45 x)$ .

$f (x) = x^{4} - \frac{25 x^{3}}{2} + \frac{95 x^{2}}{2} - (45 x) + \frac{72}{- 2}$

Step 2.6

Multiply $45$ by $- 1$ .

$f (x) = x^{4} - \frac{25 x^{3}}{2} + \frac{95 x^{2}}{2} - 45 x + \frac{72}{- 2}$

Step 2.7

Divide $72$ by $- 2$ .

$f (x) = x^{4} - \frac{25 x^{3}}{2} + \frac{95 x^{2}}{2} - 45 x - 36$

Step 3

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

$- \frac{25}{2}, \frac{95}{2}, - 45, - 36$

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.

$b_{1} = | - \frac{25}{2} |, | - 36 |, | - 45 |, | \frac{95}{2} |$

Step 4.2

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

$b_{1} = | \frac{95}{2} |$

Step 4.3

$\frac{95}{2}$ is approximately $47.5$ which is positive so remove the absolute value

$b_{1} = \frac{95}{2} + 1$

Step 4.4

Write $1$ as a fraction with a common denominator.

$b_{1} = \frac{95}{2} + \frac{2}{2}$

Step 4.5

Combine the numerators over the common denominator.

$b_{1} = \frac{95 + 2}{2}$

Step 4.6

Add $95$ and $2$ .

$b_{1} = \frac{97}{2}$

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

$- \frac{25}{2}$ is approximately $- 12.5$ which is negative so negate $- \frac{25}{2}$ and remove the absolute value

$b_{2} = \frac{25}{2} + | \frac{95}{2} | + | - 45 | + | - 36 |$

Step 5.1.2

$\frac{95}{2}$ is approximately $47.5$ which is positive so remove the absolute value

$b_{2} = \frac{25}{2} + \frac{95}{2} + | - 45 | + | - 36 |$

Step 5.1.3

The absolute value is the distance between a number and zero. The distance between $- 45$ and $0$ is $45$ .

$b_{2} = \frac{25}{2} + \frac{95}{2} + 45 + | - 36 |$

Step 5.1.4

The absolute value is the distance between a number and zero. The distance between $- 36$ and $0$ is $36$ .

$b_{2} = \frac{25}{2} + \frac{95}{2} + 45 + 36$

Step 5.2

Combine fractions.

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Step 5.2.1

Combine the numerators over the common denominator.

$b_{2} = 45 + 36 + \frac{25 + 95}{2}$

Step 5.2.2

Add $25$ and $95$ .

$b_{2} = 45 + 36 + \frac{120}{2}$

Step 5.3

Find the common denominator.

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Step 5.3.1

Write $45$ as a fraction with denominator $1$ .

$b_{2} = \frac{45}{1} + 36 + \frac{120}{2}$

Step 5.3.2

Multiply $\frac{45}{1}$ by $\frac{2}{2}$ .

$b_{2} = \frac{45}{1} \cdot \frac{2}{2} + 36 + \frac{120}{2}$

Step 5.3.3

Multiply $\frac{45}{1}$ by $\frac{2}{2}$ .

$b_{2} = \frac{45 \cdot 2}{2} + 36 + \frac{120}{2}$

Step 5.3.4

Write $36$ as a fraction with denominator $1$ .

$b_{2} = \frac{45 \cdot 2}{2} + \frac{36}{1} + \frac{120}{2}$

Step 5.3.5

Multiply $\frac{36}{1}$ by $\frac{2}{2}$ .

$b_{2} = \frac{45 \cdot 2}{2} + \frac{36}{1} \cdot \frac{2}{2} + \frac{120}{2}$

Step 5.3.6

Multiply $\frac{36}{1}$ by $\frac{2}{2}$ .

$b_{2} = \frac{45 \cdot 2}{2} + \frac{36 \cdot 2}{2} + \frac{120}{2}$

Step 5.4

Combine the numerators over the common denominator.

$b_{2} = \frac{45 \cdot 2 + 36 \cdot 2 + 120}{2}$

Step 5.5

Simplify each term.

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Step 5.5.1

Multiply $45$ by $2$ .

$b_{2} = \frac{90 + 36 \cdot 2 + 120}{2}$

Step 5.5.2

Multiply $36$ by $2$ .

$b_{2} = \frac{90 + 72 + 120}{2}$

Step 5.6

Simplify the expression.

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Step 5.6.1

Add $90$ and $72$ .

$b_{2} = \frac{162 + 120}{2}$

Step 5.6.2

Add $162$ and $120$ .

$b_{2} = \frac{282}{2}$

Step 5.6.3

Divide $282$ by $2$ .

$b_{2} = 141$

Step 5.7

Arrange the terms in ascending order.

$b_{2} = 1, 141$

Step 5.8

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

$b_{2} = 141$

Step 6

Take the smaller bound option between $b_{1} = \frac{97}{2}$ and $b_{2} = 141$ .

Smaller Bound: $\frac{97}{2}$

Step 7

Every real root on $f (x) = - 2 x^{4} + 25 x^{3} - 95 x^{2} + 90 x + 72$ lies between $- \frac{97}{2}$ and $\frac{97}{2}$ .

$- \frac{97}{2}$ and $\frac{97}{2}$