Pre-Algebra Examples

$p (x) = 6 x^{4} + 19 x^{3} + 24 x^{2} + 24 x + 8$

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: $6$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $6$ .

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Step 2.1.1

Cancel the common factor.

$p (x) = \frac{6 x^{4}}{6} + \frac{19 x^{3}}{6} + \frac{24 x^{2}}{6} + \frac{24 x}{6} + \frac{8}{6}$

Step 2.1.2

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

$p (x) = x^{4} + \frac{19 x^{3}}{6} + \frac{24 x^{2}}{6} + \frac{24 x}{6} + \frac{8}{6}$

Step 2.2

Cancel the common factor of $24$ and $6$ .

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Step 2.2.1

Factor $6$ out of $24 x^{2}$ .

$p (x) = x^{4} + \frac{19 x^{3}}{6} + \frac{6 (4 x^{2})}{6} + \frac{24 x}{6} + \frac{8}{6}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $6$ out of $6$ .

$p (x) = x^{4} + \frac{19 x^{3}}{6} + \frac{6 (4 x^{2})}{6 (1)} + \frac{24 x}{6} + \frac{8}{6}$

Step 2.2.2.2

Cancel the common factor.

$p (x) = x^{4} + \frac{19 x^{3}}{6} + \frac{6 (4 x^{2})}{6 \cdot 1} + \frac{24 x}{6} + \frac{8}{6}$

Step 2.2.2.3

Rewrite the expression.

$p (x) = x^{4} + \frac{19 x^{3}}{6} + \frac{4 x^{2}}{1} + \frac{24 x}{6} + \frac{8}{6}$

Step 2.2.2.4

Divide $4 x^{2}$ by $1$ .

$p (x) = x^{4} + \frac{19 x^{3}}{6} + 4 x^{2} + \frac{24 x}{6} + \frac{8}{6}$

Step 2.3

Cancel the common factor of $24$ and $6$ .

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Step 2.3.1

Factor $6$ out of $24 x$ .

$p (x) = x^{4} + \frac{19 x^{3}}{6} + 4 x^{2} + \frac{6 (4 x)}{6} + \frac{8}{6}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $6$ out of $6$ .

$p (x) = x^{4} + \frac{19 x^{3}}{6} + 4 x^{2} + \frac{6 (4 x)}{6 (1)} + \frac{8}{6}$

Step 2.3.2.2

Cancel the common factor.

$p (x) = x^{4} + \frac{19 x^{3}}{6} + 4 x^{2} + \frac{6 (4 x)}{6 \cdot 1} + \frac{8}{6}$

Step 2.3.2.3

Rewrite the expression.

$p (x) = x^{4} + \frac{19 x^{3}}{6} + 4 x^{2} + \frac{4 x}{1} + \frac{8}{6}$

Step 2.3.2.4

Divide $4 x$ by $1$ .

$p (x) = x^{4} + \frac{19 x^{3}}{6} + 4 x^{2} + 4 x + \frac{8}{6}$

Step 2.4

Cancel the common factor of $8$ and $6$ .

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Step 2.4.1

Factor $2$ out of $8$ .

$p (x) = x^{4} + \frac{19 x^{3}}{6} + 4 x^{2} + 4 x + \frac{2 (4)}{6}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $6$ .

$p (x) = x^{4} + \frac{19 x^{3}}{6} + 4 x^{2} + 4 x + \frac{2 \cdot 4}{2 \cdot 3}$

Step 2.4.2.2

Cancel the common factor.

$p (x) = x^{4} + \frac{19 x^{3}}{6} + 4 x^{2} + 4 x + \frac{2 \cdot 4}{2 \cdot 3}$

Step 2.4.2.3

Rewrite the expression.

$p (x) = x^{4} + \frac{19 x^{3}}{6} + 4 x^{2} + 4 x + \frac{4}{3}$

Step 3

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

$\frac{19}{6}, 4, 4, \frac{4}{3}$

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.

$b_{1} = | \frac{4}{3} |, | \frac{19}{6} |, | 4 |, | 4 |$

Step 4.2

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

$b_{1} = | 4 |$

Step 4.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$b_{1} = 4 + 1$

Step 4.4

Add $4$ and $1$ .

$b_{1} = 5$

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

$\frac{19}{6}$ is approximately $3.1\overline{6}$ which is positive so remove the absolute value

$b_{2} = \frac{19}{6} + | 4 | + | 4 | + | \frac{4}{3} |$

Step 5.1.2

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$b_{2} = \frac{19}{6} + 4 + | 4 | + | \frac{4}{3} |$

Step 5.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$b_{2} = \frac{19}{6} + 4 + 4 + | \frac{4}{3} |$

Step 5.1.4

$\frac{4}{3}$ is approximately $1.\overline{3}$ which is positive so remove the absolute value

$b_{2} = \frac{19}{6} + 4 + 4 + \frac{4}{3}$

Step 5.2

Find the common denominator.

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Step 5.2.1

Write $4$ as a fraction with denominator $1$ .

$b_{2} = \frac{19}{6} + \frac{4}{1} + 4 + \frac{4}{3}$

Step 5.2.2

Multiply $\frac{4}{1}$ by $\frac{6}{6}$ .

$b_{2} = \frac{19}{6} + \frac{4}{1} \cdot \frac{6}{6} + 4 + \frac{4}{3}$

Step 5.2.3

Multiply $\frac{4}{1}$ by $\frac{6}{6}$ .

$b_{2} = \frac{19}{6} + \frac{4 \cdot 6}{6} + 4 + \frac{4}{3}$

Step 5.2.4

Write $4$ as a fraction with denominator $1$ .

$b_{2} = \frac{19}{6} + \frac{4 \cdot 6}{6} + \frac{4}{1} + \frac{4}{3}$

Step 5.2.5

Multiply $\frac{4}{1}$ by $\frac{6}{6}$ .

$b_{2} = \frac{19}{6} + \frac{4 \cdot 6}{6} + \frac{4}{1} \cdot \frac{6}{6} + \frac{4}{3}$

Step 5.2.6

Multiply $\frac{4}{1}$ by $\frac{6}{6}$ .

$b_{2} = \frac{19}{6} + \frac{4 \cdot 6}{6} + \frac{4 \cdot 6}{6} + \frac{4}{3}$

Step 5.2.7

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$b_{2} = \frac{19}{6} + \frac{4 \cdot 6}{6} + \frac{4 \cdot 6}{6} + \frac{4}{3} \cdot \frac{2}{2}$

Step 5.2.8

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$b_{2} = \frac{19}{6} + \frac{4 \cdot 6}{6} + \frac{4 \cdot 6}{6} + \frac{4 \cdot 2}{3 \cdot 2}$

Step 5.2.9

Reorder the factors of $3 \cdot 2$ .

$b_{2} = \frac{19}{6} + \frac{4 \cdot 6}{6} + \frac{4 \cdot 6}{6} + \frac{4 \cdot 2}{2 \cdot 3}$

Step 5.2.10

Multiply $2$ by $3$ .

$b_{2} = \frac{19}{6} + \frac{4 \cdot 6}{6} + \frac{4 \cdot 6}{6} + \frac{4 \cdot 2}{6}$

Step 5.3

Combine the numerators over the common denominator.

$b_{2} = \frac{19 + 4 \cdot 6 + 4 \cdot 6 + 4 \cdot 2}{6}$

Step 5.4

Simplify each term.

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Step 5.4.1

Multiply $4$ by $6$ .

$b_{2} = \frac{19 + 24 + 4 \cdot 6 + 4 \cdot 2}{6}$

Step 5.4.2

Multiply $4$ by $6$ .

$b_{2} = \frac{19 + 24 + 24 + 4 \cdot 2}{6}$

Step 5.4.3

Multiply $4$ by $2$ .

$b_{2} = \frac{19 + 24 + 24 + 8}{6}$

Step 5.5

Reduce the expression by cancelling the common factors.

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Step 5.5.1

Add $19$ and $24$ .

$b_{2} = \frac{43 + 24 + 8}{6}$

Step 5.5.2

Simplify by adding numbers.

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Step 5.5.2.1

Add $43$ and $24$ .

$b_{2} = \frac{67 + 8}{6}$

Step 5.5.2.2

Add $67$ and $8$ .

$b_{2} = \frac{75}{6}$

Step 5.5.3

Cancel the common factor of $75$ and $6$ .

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Step 5.5.3.1

Factor $3$ out of $75$ .

$b_{2} = \frac{3 (25)}{6}$

Step 5.5.3.2

Cancel the common factors.

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Step 5.5.3.2.1

Factor $3$ out of $6$ .

$b_{2} = \frac{3 \cdot 25}{3 \cdot 2}$

Step 5.5.3.2.2

Cancel the common factor.

$b_{2} = \frac{3 \cdot 25}{3 \cdot 2}$

Step 5.5.3.2.3

Rewrite the expression.

$b_{2} = \frac{25}{2}$

Step 5.6

Arrange the terms in ascending order.

$b_{2} = 1, \frac{25}{2}$

Step 5.7

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

$b_{2} = \frac{25}{2}$

Step 6

Take the smaller bound option between $b_{1} = 5$ and $b_{2} = \frac{25}{2}$ .

Smaller Bound: $5$

Step 7

Every real root on $p (x) = 6 x^{4} + 19 x^{3} + 24 x^{2} + 24 x + 8$ lies between $- 5$ and $5$ .

$- 5$ and $5$