Pre-Algebra Examples

$- 4 x^{3} + 9 x^{2} + 3 x - 6$

Step 1

Write $- 4 x^{3} + 9 x^{2} + 3 x - 6$ as a function.

$f (x) = - 4 x^{3} + 9 x^{2} + 3 x - 6$

Step 2

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 3

Simplify each term.

Tap for more steps...

Step 3.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 3.1.1

Cancel the common factor.

$f (x) = \frac{- 4 x^{3}}{- 4} + \frac{9 x^{2}}{- 4} + \frac{3 x}{- 4} + \frac{- 6}{- 4}$

Step 3.1.2

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

$f (x) = x^{3} + \frac{9 x^{2}}{- 4} + \frac{3 x}{- 4} + \frac{- 6}{- 4}$

Step 3.2

Move the negative in front of the fraction.

$f (x) = x^{3} - \frac{9 x^{2}}{4} + \frac{3 x}{- 4} + \frac{- 6}{- 4}$

Step 3.3

Move the negative in front of the fraction.

$f (x) = x^{3} - \frac{9 x^{2}}{4} - \frac{3 x}{4} + \frac{- 6}{- 4}$

Step 3.4

Cancel the common factor of $- 6$ and $- 4$ .

Tap for more steps...

Step 3.4.1

Factor $- 2$ out of $- 6$ .

$f (x) = x^{3} - \frac{9 x^{2}}{4} - \frac{3 x}{4} + \frac{- 2 \cdot 3}{- 4}$

Step 3.4.2

Cancel the common factors.

Tap for more steps...

Step 3.4.2.1

Factor $- 2$ out of $- 4$ .

$f (x) = x^{3} - \frac{9 x^{2}}{4} - \frac{3 x}{4} + \frac{- 2 \cdot 3}{- 2 \cdot 2}$

Step 3.4.2.2

Cancel the common factor.

$f (x) = x^{3} - \frac{9 x^{2}}{4} - \frac{3 x}{4} + \frac{- 2 \cdot 3}{- 2 \cdot 2}$

Step 3.4.2.3

Rewrite the expression.

$f (x) = x^{3} - \frac{9 x^{2}}{4} - \frac{3 x}{4} + \frac{3}{2}$

Step 4

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

$- \frac{9}{4}, - \frac{3}{4}, \frac{3}{2}$

Step 5

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 $1$ .

Tap for more steps...

Step 5.1

Arrange the terms in ascending order.

$b_{1} = | - \frac{3}{4} |, | \frac{3}{2} |, | - \frac{9}{4} |$

Step 5.2

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

$b_{1} = | - \frac{9}{4} |$

Step 5.3

$- \frac{9}{4}$ is approximately $- 2.25$ which is negative so negate $- \frac{9}{4}$ and remove the absolute value

$b_{1} = \frac{9}{4} + 1$

Step 5.4

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

$b_{1} = \frac{9}{4} + \frac{4}{4}$

Step 5.5

Combine the numerators over the common denominator.

$b_{1} = \frac{9 + 4}{4}$

Step 5.6

Add $9$ and $4$ .

$b_{1} = \frac{13}{4}$

Step 6

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 $1$ .

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

$- \frac{9}{4}$ is approximately $- 2.25$ which is negative so negate $- \frac{9}{4}$ and remove the absolute value

$b_{2} = \frac{9}{4} + | - \frac{3}{4} | + | \frac{3}{2} |$

Step 6.1.2

$- \frac{3}{4}$ is approximately $- 0.75$ which is negative so negate $- \frac{3}{4}$ and remove the absolute value

$b_{2} = \frac{9}{4} + \frac{3}{4} + | \frac{3}{2} |$

Step 6.1.3

$\frac{3}{2}$ is approximately $1.5$ which is positive so remove the absolute value

$b_{2} = \frac{9}{4} + \frac{3}{4} + \frac{3}{2}$

Step 6.2

Combine fractions.

Tap for more steps...

Step 6.2.1

Combine the numerators over the common denominator.

$b_{2} = \frac{9 + 3}{4} + \frac{3}{2}$

Step 6.2.2

Simplify the expression.

Tap for more steps...

Step 6.2.2.1

Add $9$ and $3$ .

$b_{2} = \frac{12}{4} + \frac{3}{2}$

Step 6.2.2.2

Divide $12$ by $4$ .

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

Step 6.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.4

Combine $3$ and $\frac{2}{2}$ .

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

Step 6.5

Combine the numerators over the common denominator.

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

Step 6.6

Simplify the numerator.

Tap for more steps...

Step 6.6.1

Multiply $3$ by $2$ .

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

Step 6.6.2

Add $6$ and $3$ .

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

Step 6.7

Arrange the terms in ascending order.

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

Step 6.8

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

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

Step 7

Take the smaller bound option between $b_{1} = \frac{13}{4}$ and $b_{2} = \frac{9}{2}$ .

Smaller Bound: $\frac{13}{4}$

Step 8

Every real root on $f (x) = - 4 x^{3} + 9 x^{2} + 3 x - 6$ lies between $- \frac{13}{4}$ and $\frac{13}{4}$ .

$- \frac{13}{4}$ and $\frac{13}{4}$