Pre-Algebra Examples

$f (x) = 4 x (x - 7) (x + 12)$

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

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 2.1

Cancel the common factor of $4$ .

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Step 2.1.1

Cancel the common factor.

$f (x) = \frac{4 x (x - 7) (x + 12)}{4}$

Step 2.1.2

Divide $(x (x - 7)) (x + 12)$ by $1$ .

$f (x) = (x (x - 7)) (x + 12)$

Step 2.2

Apply the distributive property.

$f (x) = (x \cdot x + x \cdot - 7) (x + 12)$

Step 2.3

Multiply $x$ by $x$ .

$f (x) = (x^{2} + x \cdot - 7) (x + 12)$

Step 2.4

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

$f (x) = (x^{2} - 7 \cdot x) (x + 12)$

Step 2.5

Expand $(x^{2} - 7 x) (x + 12)$ using the FOIL Method.

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Step 2.5.1

Apply the distributive property.

$f (x) = x^{2} (x + 12) - 7 x (x + 12)$

Step 2.5.2

Apply the distributive property.

$f (x) = x^{2} x + x^{2} \cdot 12 - 7 x (x + 12)$

Step 2.5.3

Apply the distributive property.

$f (x) = x^{2} x + x^{2} \cdot 12 - 7 x \cdot x - 7 x \cdot 12$

Step 2.6

Simplify and combine like terms.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.6.1.1.1

Multiply $x^{2}$ by $x$ .

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Step 2.6.1.1.1.1

Raise $x$ to the power of $1$ .

$f (x) = x^{2} x + x^{2} \cdot 12 - 7 x \cdot x - 7 x \cdot 12$

Step 2.6.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (x) = x^{2 + 1} + x^{2} \cdot 12 - 7 x \cdot x - 7 x \cdot 12$

Step 2.6.1.1.2

Add $2$ and $1$ .

$f (x) = x^{3} + x^{2} \cdot 12 - 7 x \cdot x - 7 x \cdot 12$

Step 2.6.1.2

Move $12$ to the left of $x^{2}$ .

$f (x) = x^{3} + 12 \cdot x^{2} - 7 x \cdot x - 7 x \cdot 12$

Step 2.6.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.6.1.3.1

Move $x$ .

$f (x) = x^{3} + 12 x^{2} - 7 (x \cdot x) - 7 x \cdot 12$

Step 2.6.1.3.2

Multiply $x$ by $x$ .

$f (x) = x^{3} + 12 x^{2} - 7 x^{2} - 7 x \cdot 12$

Step 2.6.1.4

Multiply $12$ by $- 7$ .

$f (x) = x^{3} + 12 x^{2} - 7 x^{2} - 84 x$

Step 2.6.2

Subtract $7 x^{2}$ from $12 x^{2}$ .

$f (x) = x^{3} + 5 x^{2} - 84 x$

Step 3

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

$5, - 84$

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.

$b_{1} = | 5 |, | - 84 |$

Step 4.2

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

$b_{1} = | - 84 |$

Step 4.3

The absolute value is the distance between a number and zero. The distance between $- 84$ and $0$ is $84$ .

$b_{1} = 84 + 1$

Step 4.4

Add $84$ and $1$ .

$b_{1} = 85$

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

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$b_{2} = 5 + | - 84 |$

Step 5.1.2

The absolute value is the distance between a number and zero. The distance between $- 84$ and $0$ is $84$ .

$b_{2} = 5 + 84$

Step 5.2

Add $5$ and $84$ .

$b_{2} = 89$

Step 5.3

Arrange the terms in ascending order.

$b_{2} = 1, 89$

Step 5.4

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

$b_{2} = 89$

Step 6

Take the smaller bound option between $b_{1} = 85$ and $b_{2} = 89$ .

Smaller Bound: $85$

Step 7

Every real root on $f (x) = 4 x (x - 7) (x + 12)$ lies between $- 85$ and $85$ .

$- 85$ and $85$