Pre-Algebra Examples

$f (x) = (x^{4} - 6 x^{3} + 5 x^{2} - x + 3) - (- 4 x^{3} + 2 x^{2} + 3 x + 99)$

Step 1

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

Largest Degree: $4$

Leading Coefficient: $1$

Step 2

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

$- 1$

Step 3

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 3.1

Arrange the terms in ascending order.

$b_{1} = | - 1 |$

Step 3.2

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

$b_{1} = 1 + 1$

Step 3.3

Add $1$ and $1$ .

$b_{1} = 2$

Step 4

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 4.1

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

$b_{2} = 1$

Step 4.2

Arrange the terms in ascending order.

$b_{2} = 1, 1$

Step 4.3

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

$b_{2} = 1$

Step 5

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

Smaller Bound: $1$

Step 6

Every real root on $f (x) = (x^{4} - 6 x^{3} + 5 x^{2} - x + 3) - (- 4 x^{3} + 2 x^{2} + 3 x + 99)$ lies between $- 1$ and $1$ .

$- 1$ and $1$