Pre-Algebra Examples

$h (x) = - 3 x^{2} + 4 x - 4$

Step 1

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 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.

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

Step 2.1.2

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

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

Step 2.2

Move the negative in front of the fraction.

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

Step 2.3

Dividing two negative values results in a positive value.

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

Step 3

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

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

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.

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

Step 4.2

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

$b_{1} = | \frac{4}{3} |$

Step 4.3

$\frac{4}{3}$ is approximately $1.\overline{3}$ which is positive so remove the absolute value

$b_{1} = \frac{4}{3} + 1$

Step 4.4

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

$b_{1} = \frac{4}{3} + \frac{3}{3}$

Step 4.5

Combine the numerators over the common denominator.

$b_{1} = \frac{4 + 3}{3}$

Step 4.6

Add $4$ and $3$ .

$b_{1} = \frac{7}{3}$

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

$- \frac{4}{3}$ is approximately $- 1.\overline{3}$ which is negative so negate $- \frac{4}{3}$ and remove the absolute value

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

Step 5.1.2

$\frac{4}{3}$ is approximately $1.\overline{3}$ which is positive so remove the absolute value

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

Step 5.2

Combine fractions.

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Step 5.2.1

Combine the numerators over the common denominator.

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

Step 5.2.2

Add $4$ and $4$ .

$b_{2} = \frac{8}{3}$

Step 5.3

Arrange the terms in ascending order.

$b_{2} = 1, \frac{8}{3}$

Step 5.4

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

$b_{2} = \frac{8}{3}$

Step 6

Take the smaller bound option between $b_{1} = \frac{7}{3}$ and $b_{2} = \frac{8}{3}$ .

Smaller Bound: $\frac{7}{3}$

Step 7

Every real root on $h (x) = - 3 x^{2} + 4 x - 4$ lies between $- \frac{7}{3}$ and $\frac{7}{3}$ .

$- \frac{7}{3}$ and $\frac{7}{3}$