Pre-Algebra Examples

$f (x) = - 2 (x - 7) {(x + 9)}^{2}$

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: $- 2$

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 $- 2$ .

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Step 2.1.1

Cancel the common factor.

$f (x) = \frac{- 2 (x - 7) {(x + 9)}^{2}}{- 2}$

Step 2.1.2

Divide $(x - 7) {(x + 9)}^{2}$ by $1$ .

$f (x) = (x - 7) {(x + 9)}^{2}$

Step 2.2

Rewrite ${(x + 9)}^{2}$ as $(x + 9) (x + 9)$ .

$f (x) = (x - 7) ((x + 9) (x + 9))$

Step 2.3

Expand $(x + 9) (x + 9)$ using the FOIL Method.

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Step 2.3.1

Apply the distributive property.

$f (x) = (x - 7) (x (x + 9) + 9 (x + 9))$

Step 2.3.2

Apply the distributive property.

$f (x) = (x - 7) (x \cdot x + x \cdot 9 + 9 (x + 9))$

Step 2.3.3

Apply the distributive property.

$f (x) = (x - 7) (x \cdot x + x \cdot 9 + 9 x + 9 \cdot 9)$

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Multiply $x$ by $x$ .

$f (x) = (x - 7) (x^{2} + x \cdot 9 + 9 x + 9 \cdot 9)$

Step 2.4.1.2

Move $9$ to the left of $x$ .

$f (x) = (x - 7) (x^{2} + 9 \cdot x + 9 x + 9 \cdot 9)$

Step 2.4.1.3

Multiply $9$ by $9$ .

$f (x) = (x - 7) (x^{2} + 9 x + 9 x + 81)$

Step 2.4.2

Add $9 x$ and $9 x$ .

$f (x) = (x - 7) (x^{2} + 18 x + 81)$

Step 2.5

Expand $(x - 7) (x^{2} + 18 x + 81)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = x \cdot x^{2} + x (18 x) + x \cdot 81 - 7 x^{2} - 7 (18 x) - 7 \cdot 81$

Step 2.6

Simplify each term.

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Step 2.6.1

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

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Step 2.6.1.1

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

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Step 2.6.1.1.1

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

$f (x) = x \cdot x^{2} + x (18 x) + x \cdot 81 - 7 x^{2} - 7 (18 x) - 7 \cdot 81$

Step 2.6.1.1.2

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

$f (x) = x^{1 + 2} + x (18 x) + x \cdot 81 - 7 x^{2} - 7 (18 x) - 7 \cdot 81$

Step 2.6.1.2

Add $1$ and $2$ .

$f (x) = x^{3} + x (18 x) + x \cdot 81 - 7 x^{2} - 7 (18 x) - 7 \cdot 81$

Step 2.6.2

Rewrite using the commutative property of multiplication.

$f (x) = x^{3} + 18 x \cdot x + x \cdot 81 - 7 x^{2} - 7 (18 x) - 7 \cdot 81$

Step 2.6.3

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

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Step 2.6.3.1

Move $x$ .

$f (x) = x^{3} + 18 (x \cdot x) + x \cdot 81 - 7 x^{2} - 7 (18 x) - 7 \cdot 81$

Step 2.6.3.2

Multiply $x$ by $x$ .

$f (x) = x^{3} + 18 x^{2} + x \cdot 81 - 7 x^{2} - 7 (18 x) - 7 \cdot 81$

Step 2.6.4

Move $81$ to the left of $x$ .

$f (x) = x^{3} + 18 x^{2} + 81 \cdot x - 7 x^{2} - 7 (18 x) - 7 \cdot 81$

Step 2.6.5

Multiply $18$ by $- 7$ .

$f (x) = x^{3} + 18 x^{2} + 81 x - 7 x^{2} - 126 x - 7 \cdot 81$

Step 2.6.6

Multiply $- 7$ by $81$ .

$f (x) = x^{3} + 18 x^{2} + 81 x - 7 x^{2} - 126 x - 567$

Step 2.7

Subtract $7 x^{2}$ from $18 x^{2}$ .

$f (x) = x^{3} + 11 x^{2} + 81 x - 126 x - 567$

Step 2.8

Subtract $126 x$ from $81 x$ .

$f (x) = x^{3} + 11 x^{2} - 45 x - 567$

Step 3

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

$11, - 45, - 567$

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.

$b_{1} = | 11 |, | - 45 |, | - 567 |$

Step 4.2

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

$b_{1} = | - 567 |$

Step 4.3

The absolute value is the distance between a number and zero. The distance between $- 567$ and $0$ is $567$ .

$b_{1} = 567 + 1$

Step 4.4

Add $567$ and $1$ .

$b_{1} = 568$

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 $11$ is $11$ .

$b_{2} = 11 + | - 45 | + | - 567 |$

Step 5.1.2

The absolute value is the distance between a number and zero. The distance between $- 45$ and $0$ is $45$ .

$b_{2} = 11 + 45 + | - 567 |$

Step 5.1.3

The absolute value is the distance between a number and zero. The distance between $- 567$ and $0$ is $567$ .

$b_{2} = 11 + 45 + 567$

Step 5.2

Simplify by adding numbers.

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Step 5.2.1

Add $11$ and $45$ .

$b_{2} = 56 + 567$

Step 5.2.2

Add $56$ and $567$ .

$b_{2} = 623$

Step 5.3

Arrange the terms in ascending order.

$b_{2} = 1, 623$

Step 5.4

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

$b_{2} = 623$

Step 6

Take the smaller bound option between $b_{1} = 568$ and $b_{2} = 623$ .

Smaller Bound: $568$

Step 7

Every real root on $f (x) = - 2 (x - 7) {(x + 9)}^{2}$ lies between $- 568$ and $568$ .

$- 568$ and $568$