Pre-Algebra Examples

$3 - 2 \sqrt{5}$

Step 1

Write $3 - 2 \sqrt{5}$ as a function.

$f (x) = 3 - 2 \sqrt{5}$

Step 2

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

Largest Degree: $0$

Leading Coefficient: $3$

Step 3

Simplify each term.

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Step 3.1

Divide $3$ by $3$ .

$f (x) = 1 + \frac{- 2 \sqrt{5}}{3}$

Step 3.2

Move the negative in front of the fraction.

$f (x) = 1 - \frac{2 \sqrt{5}}{3}$

Step 4

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

Largest Degree: $0$

Leading Coefficient: $- 2$

Step 5

Simplify each term.

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Step 5.1

Move the negative in front of the fraction.

$f (x) = - \frac{1}{2} + \frac{- \frac{2 \sqrt{5}}{3}}{- 2}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$f (x) = - \frac{1}{2} - \frac{2 \sqrt{5}}{3} \cdot \frac{1}{- 2}$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Move the leading negative in $- \frac{2 \sqrt{5}}{3}$ into the numerator.

$f (x) = - \frac{1}{2} + \frac{- 2 \sqrt{5}}{3} \cdot \frac{1}{- 2}$

Step 5.3.2

Factor $2$ out of $- 2 \sqrt{5}$ .

$f (x) = - \frac{1}{2} + \frac{2 (- \sqrt{5})}{3} \cdot \frac{1}{- 2}$

Step 5.3.3

Factor $2$ out of $- 2$ .

$f (x) = - \frac{1}{2} + \frac{2 (- \sqrt{5})}{3} \cdot \frac{1}{2 (- 1)}$

Step 5.3.4

Cancel the common factor.

$f (x) = - \frac{1}{2} + \frac{2 (- \sqrt{5})}{3} \cdot \frac{1}{2 \cdot - 1}$

Step 5.3.5

Rewrite the expression.

$f (x) = - \frac{1}{2} + \frac{- \sqrt{5}}{3} \cdot \frac{1}{- 1}$

Step 5.4

Multiply $\frac{- \sqrt{5}}{3}$ by $\frac{1}{- 1}$ .

$f (x) = - \frac{1}{2} + \frac{- \sqrt{5}}{3 \cdot - 1}$

Step 5.5

Multiply $3$ by $- 1$ .

$f (x) = - \frac{1}{2} + \frac{- \sqrt{5}}{- 3}$

Step 5.6

Dividing two negative values results in a positive value.

$f (x) = - \frac{1}{2} + \frac{\sqrt{5}}{3}$

Step 6

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

$0$

Step 7

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 7.1

Arrange the terms in ascending order.

$b_{1} = 0$

Step 7.2

Add $0$ and $1$ .

$b_{1} = 1$

Step 8

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 8.1

Arrange the terms in ascending order.

$b_{2} = 0, 1$

Step 8.2

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

$b_{2} = 1$

Step 9

The bound options are the same.

Bound: $1$

Step 10

Every real root on $f (x) = 3 - 2 \sqrt{5}$ lies between $- 1$ and $1$ .

$- 1$ and $1$