Pre-Algebra Examples

$f (x) = 3 x^{3} + x^{2} - 4 x - 12$

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: $3$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $3$ .

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Step 2.1.1

Cancel the common factor.

$f (x) = \frac{3 x^{3}}{3} + \frac{x^{2}}{3} + \frac{- 4 x}{3} + \frac{- 12}{3}$

Step 2.1.2

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

$f (x) = x^{3} + \frac{x^{2}}{3} + \frac{- 4 x}{3} + \frac{- 12}{3}$

Step 2.2

Move the negative in front of the fraction.

$f (x) = x^{3} + \frac{x^{2}}{3} - \frac{4 x}{3} + \frac{- 12}{3}$

Step 2.3

Divide $- 12$ by $3$ .

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

Step 3

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

$\frac{1}{3}, - \frac{4}{3}, - 4$

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.

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

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 $- 4$ and $0$ 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

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$b_{2} = \frac{1}{3} + | - \frac{4}{3} | + | - 4 |$

Step 5.1.2

$- \frac{4}{3}$ is approximately $- 1.\overline{3}$ which is negative so negate $- \frac{4}{3}$ and remove the absolute value

$b_{2} = \frac{1}{3} + \frac{4}{3} + | - 4 |$

Step 5.1.3

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

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

Step 5.2

Combine fractions.

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Step 5.2.1

Combine the numerators over the common denominator.

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

Step 5.2.2

Add $1$ and $4$ .

$b_{2} = 4 + \frac{5}{3}$

Step 5.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$b_{2} = 4 \cdot \frac{3}{3} + \frac{5}{3}$

Step 5.4

Combine $4$ and $\frac{3}{3}$ .

$b_{2} = \frac{4 \cdot 3}{3} + \frac{5}{3}$

Step 5.5

Combine the numerators over the common denominator.

$b_{2} = \frac{4 \cdot 3 + 5}{3}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $4$ by $3$ .

$b_{2} = \frac{12 + 5}{3}$

Step 5.6.2

Add $12$ and $5$ .

$b_{2} = \frac{17}{3}$

Step 5.7

Arrange the terms in ascending order.

$b_{2} = 1, \frac{17}{3}$

Step 5.8

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

$b_{2} = \frac{17}{3}$

Step 6

Take the smaller bound option between $b_{1} = 5$ and $b_{2} = \frac{17}{3}$ .

Smaller Bound: $5$

Step 7

Every real root on $f (x) = 3 x^{3} + x^{2} - 4 x - 12$ lies between $- 5$ and $5$ .

$- 5$ and $5$