Pre-Algebra Examples

$f (x) = 6 x^{2} \cdot (7 x) - 3$

Step 1

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

$6, 1, - 3$

Step 2

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 2.1

Arrange the terms in ascending order.

$b_{1} = | 1 |, | - 3 |, | 6 |$

Step 2.2

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

$b_{1} = | 6 |$

Step 2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$b_{1} = 6 + 1$

Step 2.4

Add $6$ and $1$ .

$b_{1} = 7$

Step 3

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 3.1

Simplify each term.

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Step 3.1.1

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$b_{2} = 6 + | 1 | + | - 3 |$

Step 3.1.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$b_{2} = 6 + 1 + | - 3 |$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $- 3$ and $0$ is $3$ .

$b_{2} = 6 + 1 + 3$

Step 3.2

Simplify by adding numbers.

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Step 3.2.1

Add $6$ and $1$ .

$b_{2} = 7 + 3$

Step 3.2.2

Add $7$ and $3$ .

$b_{2} = 10$

Step 3.3

Arrange the terms in ascending order.

$b_{2} = 1, 10$

Step 3.4

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

$b_{2} = 10$

Step 4

Take the smaller bound option between $b_{1} = 7$ and $b_{2} = 10$ .

Smaller Bound: $7$

Step 5

Every real root on $f (x) = 6 x^{2} \cdot (7 x) - 3$ lies between $- 7$ and $7$ .

$- 7$ and $7$