Pre-Algebra Examples

$f (x) = 3 x^{3} + 39 x^{2} + 39 x + 27$

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.

$f (x) = \frac{3 x^{3}}{3} + \frac{39 x^{2}}{3} + \frac{39 x}{3} + \frac{27}{3}$

Step 2.1.2

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

$f (x) = x^{3} + \frac{39 x^{2}}{3} + \frac{39 x}{3} + \frac{27}{3}$

Step 2.2

Cancel the common factor of $39$ and $3$ .

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Step 2.2.1

Factor $3$ out of $39 x^{2}$ .

$f (x) = x^{3} + \frac{3 (13 x^{2})}{3} + \frac{39 x}{3} + \frac{27}{3}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $3$ out of $3$ .

$f (x) = x^{3} + \frac{3 (13 x^{2})}{3 (1)} + \frac{39 x}{3} + \frac{27}{3}$

Step 2.2.2.2

Cancel the common factor.

$f (x) = x^{3} + \frac{3 (13 x^{2})}{3 \cdot 1} + \frac{39 x}{3} + \frac{27}{3}$

Step 2.2.2.3

Rewrite the expression.

$f (x) = x^{3} + \frac{13 x^{2}}{1} + \frac{39 x}{3} + \frac{27}{3}$

Step 2.2.2.4

Divide $13 x^{2}$ by $1$ .

$f (x) = x^{3} + 13 x^{2} + \frac{39 x}{3} + \frac{27}{3}$

Step 2.3

Cancel the common factor of $39$ and $3$ .

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Step 2.3.1

Factor $3$ out of $39 x$ .

$f (x) = x^{3} + 13 x^{2} + \frac{3 (13 x)}{3} + \frac{27}{3}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $3$ out of $3$ .

$f (x) = x^{3} + 13 x^{2} + \frac{3 (13 x)}{3 (1)} + \frac{27}{3}$

Step 2.3.2.2

Cancel the common factor.

$f (x) = x^{3} + 13 x^{2} + \frac{3 (13 x)}{3 \cdot 1} + \frac{27}{3}$

Step 2.3.2.3

Rewrite the expression.

$f (x) = x^{3} + 13 x^{2} + \frac{13 x}{1} + \frac{27}{3}$

Step 2.3.2.4

Divide $13 x$ by $1$ .

$f (x) = x^{3} + 13 x^{2} + 13 x + \frac{27}{3}$

Step 2.4

Divide $27$ by $3$ .

$f (x) = x^{3} + 13 x^{2} + 13 x + 9$

Step 3

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

$13, 13, 9$

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.

$b_{1} = | 9 |, | 13 |, | 13 |$

Step 4.2

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

$b_{1} = | 13 |$

Step 4.3

The absolute value is the distance between a number and zero. The distance between $0$ and $13$ is $13$ .

$b_{1} = 13 + 1$

Step 4.4

Add $13$ and $1$ .

$b_{1} = 14$

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

The absolute value is the distance between a number and zero. The distance between $0$ and $13$ is $13$ .

$b_{2} = 13 + | 13 | + | 9 |$

Step 5.1.2

The absolute value is the distance between a number and zero. The distance between $0$ and $13$ is $13$ .

$b_{2} = 13 + 13 + | 9 |$

Step 5.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $9$ is $9$ .

$b_{2} = 13 + 13 + 9$

Step 5.2

Simplify by adding numbers.

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Step 5.2.1

Add $13$ and $13$ .

$b_{2} = 26 + 9$

Step 5.2.2

Add $26$ and $9$ .

$b_{2} = 35$

Step 5.3

Arrange the terms in ascending order.

$b_{2} = 1, 35$

Step 5.4

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

$b_{2} = 35$

Step 6

Take the smaller bound option between $b_{1} = 14$ and $b_{2} = 35$ .

Smaller Bound: $14$

Step 7

Every real root on $f (x) = 3 x^{3} + 39 x^{2} + 39 x + 27$ lies between $- 14$ and $14$ .

$- 14$ and $14$