Pre-Algebra Examples

$f (x) = 4 x^{3} - x^{2} - 256 x + 64$

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: $4$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $4$ .

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Step 2.1.1

Cancel the common factor.

$f (x) = \frac{4 x^{3}}{4} + \frac{- x^{2}}{4} + \frac{- 256 x}{4} + \frac{64}{4}$

Step 2.1.2

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

$f (x) = x^{3} + \frac{- x^{2}}{4} + \frac{- 256 x}{4} + \frac{64}{4}$

Step 2.2

Move the negative in front of the fraction.

$f (x) = x^{3} - \frac{x^{2}}{4} + \frac{- 256 x}{4} + \frac{64}{4}$

Step 2.3

Cancel the common factor of $- 256$ and $4$ .

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Step 2.3.1

Factor $4$ out of $- 256 x$ .

$f (x) = x^{3} - \frac{x^{2}}{4} + \frac{4 (- 64 x)}{4} + \frac{64}{4}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $4$ out of $4$ .

$f (x) = x^{3} - \frac{x^{2}}{4} + \frac{4 (- 64 x)}{4 (1)} + \frac{64}{4}$

Step 2.3.2.2

Cancel the common factor.

$f (x) = x^{3} - \frac{x^{2}}{4} + \frac{4 (- 64 x)}{4 \cdot 1} + \frac{64}{4}$

Step 2.3.2.3

Rewrite the expression.

$f (x) = x^{3} - \frac{x^{2}}{4} + \frac{- 64 x}{1} + \frac{64}{4}$

Step 2.3.2.4

Divide $- 64 x$ by $1$ .

$f (x) = x^{3} - \frac{x^{2}}{4} - 64 x + \frac{64}{4}$

Step 2.4

Divide $64$ by $4$ .

$f (x) = x^{3} - \frac{x^{2}}{4} - 64 x + 16$

Step 3

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

$- \frac{1}{4}, - 64, 16$

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.

$b_{1} = | - \frac{1}{4} |, | 16 |, | - 64 |$

Step 4.2

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

$b_{1} = | - 64 |$

Step 4.3

The absolute value is the distance between a number and zero. The distance between $- 64$ and $0$ is $64$ .

$b_{1} = 64 + 1$

Step 4.4

Add $64$ and $1$ .

$b_{1} = 65$

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

$- \frac{1}{4}$ is approximately $- 0.25$ which is negative so negate $- \frac{1}{4}$ and remove the absolute value

$b_{2} = \frac{1}{4} + | - 64 | + | 16 |$

Step 5.1.2

The absolute value is the distance between a number and zero. The distance between $- 64$ and $0$ is $64$ .

$b_{2} = \frac{1}{4} + 64 + | 16 |$

Step 5.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $16$ is $16$ .

$b_{2} = \frac{1}{4} + 64 + 16$

Step 5.2

Find the common denominator.

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Step 5.2.1

Write $64$ as a fraction with denominator $1$ .

$b_{2} = \frac{1}{4} + \frac{64}{1} + 16$

Step 5.2.2

Multiply $\frac{64}{1}$ by $\frac{4}{4}$ .

$b_{2} = \frac{1}{4} + \frac{64}{1} \cdot \frac{4}{4} + 16$

Step 5.2.3

Multiply $\frac{64}{1}$ by $\frac{4}{4}$ .

$b_{2} = \frac{1}{4} + \frac{64 \cdot 4}{4} + 16$

Step 5.2.4

Write $16$ as a fraction with denominator $1$ .

$b_{2} = \frac{1}{4} + \frac{64 \cdot 4}{4} + \frac{16}{1}$

Step 5.2.5

Multiply $\frac{16}{1}$ by $\frac{4}{4}$ .

$b_{2} = \frac{1}{4} + \frac{64 \cdot 4}{4} + \frac{16}{1} \cdot \frac{4}{4}$

Step 5.2.6

Multiply $\frac{16}{1}$ by $\frac{4}{4}$ .

$b_{2} = \frac{1}{4} + \frac{64 \cdot 4}{4} + \frac{16 \cdot 4}{4}$

Step 5.3

Combine the numerators over the common denominator.

$b_{2} = \frac{1 + 64 \cdot 4 + 16 \cdot 4}{4}$

Step 5.4

Simplify each term.

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Step 5.4.1

Multiply $64$ by $4$ .

$b_{2} = \frac{1 + 256 + 16 \cdot 4}{4}$

Step 5.4.2

Multiply $16$ by $4$ .

$b_{2} = \frac{1 + 256 + 64}{4}$

Step 5.5

Simplify by adding numbers.

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Step 5.5.1

Add $1$ and $256$ .

$b_{2} = \frac{257 + 64}{4}$

Step 5.5.2

Add $257$ and $64$ .

$b_{2} = \frac{321}{4}$

Step 5.6

Arrange the terms in ascending order.

$b_{2} = 1, \frac{321}{4}$

Step 5.7

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

$b_{2} = \frac{321}{4}$

Step 6

Take the smaller bound option between $b_{1} = 65$ and $b_{2} = \frac{321}{4}$ .

Smaller Bound: $65$

Step 7

Every real root on $f (x) = 4 x^{3} - x^{2} - 256 x + 64$ lies between $- 65$ and $65$ .

$- 65$ and $65$