Pre-Algebra Examples

$f (x) = 4 x^{3} + x^{2} + 10 x - 14$

Step 1

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

Largest Degree: $3$

Leading Coefficient: $4$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $4$ .

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Step 2.1.1

Cancel the common factor.

$f (x) = \frac{4 x^{3}}{4} + \frac{x^{2}}{4} + \frac{10 x}{4} + \frac{- 14}{4}$

Step 2.1.2

Divide $x^{3}$ by $1$ .

$f (x) = x^{3} + \frac{x^{2}}{4} + \frac{10 x}{4} + \frac{- 14}{4}$

Step 2.2

Cancel the common factor of $10$ and $4$ .

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Step 2.2.1

Factor $2$ out of $10 x$ .

$f (x) = x^{3} + \frac{x^{2}}{4} + \frac{2 (5 x)}{4} + \frac{- 14}{4}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $2$ out of $4$ .

$f (x) = x^{3} + \frac{x^{2}}{4} + \frac{2 (5 x)}{2 (2)} + \frac{- 14}{4}$

Step 2.2.2.2

Cancel the common factor.

$f (x) = x^{3} + \frac{x^{2}}{4} + \frac{2 (5 x)}{2 \cdot 2} + \frac{- 14}{4}$

Step 2.2.2.3

Rewrite the expression.

$f (x) = x^{3} + \frac{x^{2}}{4} + \frac{5 x}{2} + \frac{- 14}{4}$

Step 2.3

Cancel the common factor of $- 14$ and $4$ .

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Step 2.3.1

Factor $2$ out of $- 14$ .

$f (x) = x^{3} + \frac{x^{2}}{4} + \frac{5 x}{2} + \frac{2 (- 7)}{4}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $4$ .

$f (x) = x^{3} + \frac{x^{2}}{4} + \frac{5 x}{2} + \frac{2 \cdot - 7}{2 \cdot 2}$

Step 2.3.2.2

Cancel the common factor.

$f (x) = x^{3} + \frac{x^{2}}{4} + \frac{5 x}{2} + \frac{2 \cdot - 7}{2 \cdot 2}$

Step 2.3.2.3

Rewrite the expression.

$f (x) = x^{3} + \frac{x^{2}}{4} + \frac{5 x}{2} + \frac{- 7}{2}$

Step 2.4

Move the negative in front of the fraction.

$f (x) = x^{3} + \frac{x^{2}}{4} + \frac{5 x}{2} - \frac{7}{2}$

Step 3

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

$\frac{1}{4}, \frac{5}{2}, - \frac{7}{2}$

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.

$b_{1} = | \frac{1}{4} |, | \frac{5}{2} |, | - \frac{7}{2} |$

Step 4.2

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

$b_{1} = | - \frac{7}{2} |$

Step 4.3

$- \frac{7}{2}$ is approximately $- 3.5$ which is negative so negate $- \frac{7}{2}$ and remove the absolute value

$b_{1} = \frac{7}{2} + 1$

Step 4.4

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

$b_{1} = \frac{7}{2} + \frac{2}{2}$

Step 4.5

Combine the numerators over the common denominator.

$b_{1} = \frac{7 + 2}{2}$

Step 4.6

Add $7$ and $2$ .

$b_{1} = \frac{9}{2}$

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

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$b_{2} = \frac{1}{4} + | \frac{5}{2} | + | - \frac{7}{2} |$

Step 5.1.2

$\frac{5}{2}$ is approximately $2.5$ which is positive so remove the absolute value

$b_{2} = \frac{1}{4} + \frac{5}{2} + | - \frac{7}{2} |$

Step 5.1.3

$- \frac{7}{2}$ is approximately $- 3.5$ which is negative so negate $- \frac{7}{2}$ and remove the absolute value

$b_{2} = \frac{1}{4} + \frac{5}{2} + \frac{7}{2}$

Step 5.2

Combine fractions.

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Step 5.2.1

Combine the numerators over the common denominator.

$b_{2} = \frac{1}{4} + \frac{5 + 7}{2}$

Step 5.2.2

Simplify the expression.

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Step 5.2.2.1

Add $5$ and $7$ .

$b_{2} = \frac{1}{4} + \frac{12}{2}$

Step 5.2.2.2

Divide $12$ by $2$ .

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

Step 5.3

To write $6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 5.4

Combine $6$ and $\frac{4}{4}$ .

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

Step 5.5

Combine the numerators over the common denominator.

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

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $6$ by $4$ .

$b_{2} = \frac{1 + 24}{4}$

Step 5.6.2

Add $1$ and $24$ .

$b_{2} = \frac{25}{4}$

Step 5.7

Arrange the terms in ascending order.

$b_{2} = 1, \frac{25}{4}$

Step 5.8

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

$b_{2} = \frac{25}{4}$

Step 6

Take the smaller bound option between $b_{1} = \frac{9}{2}$ and $b_{2} = \frac{25}{4}$ .

Smaller Bound: $\frac{9}{2}$

Step 7

Every real root on $f (x) = 4 x^{3} + x^{2} + 10 x - 14$ lies between $- \frac{9}{2}$ and $\frac{9}{2}$ .

$- \frac{9}{2}$ and $\frac{9}{2}$