Pre-Algebra Examples

$3 x (x + 7)$

Step 1

Write $3 x (x + 7)$ as a function.

$f (x) = 3 x (x + 7)$

Step 2

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

Largest Degree: $2$

Leading Coefficient: $3$

Step 3

The leading coefficient needs to be $1$ . If it is not, divide the expression by it to make it $1$ .

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Step 3.1

Cancel the common factor of $3$ .

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Step 3.1.1

Cancel the common factor.

$f (x) = \frac{3 x (x + 7)}{3}$

Step 3.1.2

Divide $x (x + 7)$ by $1$ .

$f (x) = x (x + 7)$

Step 3.2

Apply the distributive property.

$f (x) = x \cdot x + x \cdot 7$

Step 3.3

Multiply $x$ by $x$ .

$f (x) = x^{2} + x \cdot 7$

Step 3.4

Move $7$ to the left of $x$ .

$f (x) = x^{2} + 7 x$

Step 4

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

$7$

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 $1$ .

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Step 5.1

Arrange the terms in ascending order.

$b_{1} = | 7 |$

Step 5.2

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$b_{1} = 7 + 1$

Step 5.3

Add $7$ and $1$ .

$b_{1} = 8$

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 $1$ .

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Step 6.1

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$b_{2} = 7$

Step 6.2

Arrange the terms in ascending order.

$b_{2} = 1, 7$

Step 6.3

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

$b_{2} = 7$

Step 7

Take the smaller bound option between $b_{1} = 8$ and $b_{2} = 7$ .

Smaller Bound: $7$

Step 8

Every real root on $f (x) = 3 x (x + 7)$ lies between $- 7$ and $7$ .

$- 7$ and $7$