Pre-Algebra Examples

$p (x) = 2.75 x^{4} - 3 x^{2} + 5 x + 2$

Step 1

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

Largest Degree: $4$

Leading Coefficient: $2.75$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $2.75$ .

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Step 2.1.1

Cancel the common factor.

$p (x) = \frac{2.75 x^{4}}{2.75} + \frac{- 3 x^{2}}{2.75} + \frac{5 x}{2.75} + \frac{2}{2.75}$

Step 2.1.2

Divide $x^{4}$ by $1$ .

$p (x) = x^{4} + \frac{- 3 x^{2}}{2.75} + \frac{5 x}{2.75} + \frac{2}{2.75}$

Step 2.2

Move the negative in front of the fraction.

$p (x) = x^{4} - \frac{3 x^{2}}{2.75} + \frac{5 x}{2.75} + \frac{2}{2.75}$

Step 2.3

Factor $3$ out of $3 x^{2}$ .

$p (x) = x^{4} - \frac{3 (x^{2})}{2.75} + \frac{5 x}{2.75} + \frac{2}{2.75}$

Step 2.4

Factor $2.75$ out of $2.75$ .

$p (x) = x^{4} - \frac{3 (x^{2})}{2.75 (1)} + \frac{5 x}{2.75} + \frac{2}{2.75}$

Step 2.5

Separate fractions.

$p (x) = x^{4} - (\frac{3}{2.75} \cdot \frac{x^{2}}{1}) + \frac{5 x}{2.75} + \frac{2}{2.75}$

Step 2.6

Divide $3$ by $2.75$ .

$p (x) = x^{4} - (1.\overline{09} (\frac{x^{2}}{1})) + \frac{5 x}{2.75} + \frac{2}{2.75}$

Step 2.7

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

$p (x) = x^{4} - (1.\overline{09} x^{2}) + \frac{5 x}{2.75} + \frac{2}{2.75}$

Step 2.8

Multiply $1.\overline{09}$ by $- 1$ .

$p (x) = x^{4} - 1.\overline{09} x^{2} + \frac{5 x}{2.75} + \frac{2}{2.75}$

Step 2.9

Factor $5$ out of $5 x$ .

$p (x) = x^{4} - 1.\overline{09} x^{2} + \frac{5 (x)}{2.75} + \frac{2}{2.75}$

Step 2.10

Factor $2.75$ out of $2.75$ .

$p (x) = x^{4} - 1.\overline{09} x^{2} + \frac{5 (x)}{2.75 (1)} + \frac{2}{2.75}$

Step 2.11

Separate fractions.

$p (x) = x^{4} - 1.\overline{09} x^{2} + \frac{5}{2.75} \cdot \frac{x}{1} + \frac{2}{2.75}$

Step 2.12

Divide $5$ by $2.75$ .

$p (x) = x^{4} - 1.\overline{09} x^{2} + 1.\overline{81} (\frac{x}{1}) + \frac{2}{2.75}$

Step 2.13

Divide $x$ by $1$ .

$p (x) = x^{4} - 1.\overline{09} x^{2} + 1.\overline{81} x + \frac{2}{2.75}$

Step 2.14

Divide $2$ by $2.75$ .

$p (x) = x^{4} - 1.\overline{09} x^{2} + 1.\overline{81} x + 0.\overline{72}$

Step 3

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

$- 1.\overline{09}, 1.\overline{81}, 0.\overline{72}$

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.

$b_{1} = | 0.\overline{72} |, | - 1.\overline{09} |, | 1.\overline{81} |$

Step 4.2

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

$b_{1} = | 1.\overline{81} |$

Step 4.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1.\overline{81}$ is $1.\overline{81}$ .

$b_{1} = 1.\overline{81} + 1$

Step 4.4

Add $1.\overline{81}$ and $1$ .

$b_{1} = 2.\overline{81}$

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 $- 1.\overline{09}$ and $0$ is $1.\overline{09}$ .

$b_{2} = 1.\overline{09} + | 1.\overline{81} | + | 0.\overline{72} |$

Step 5.1.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1.\overline{81}$ is $1.\overline{81}$ .

$b_{2} = 1.\overline{09} + 1.\overline{81} + | 0.\overline{72} |$

Step 5.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0.\overline{72}$ is $0.\overline{72}$ .

$b_{2} = 1.\overline{09} + 1.\overline{81} + 0.\overline{72}$

Step 5.2

Simplify by adding numbers.

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Step 5.2.1

Add $1.\overline{09}$ and $1.\overline{81}$ .

$b_{2} = 2.\overline{90} + 0.\overline{72}$

Step 5.2.2

Add $2.\overline{90}$ and $0.\overline{72}$ .

$b_{2} = 3.\overline{63}$

Step 5.3

Arrange the terms in ascending order.

$b_{2} = 1, 3.\overline{63}$

Step 5.4

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

$b_{2} = 3.\overline{63}$

Step 6

Take the smaller bound option between $b_{1} = 2.\overline{81}$ and $b_{2} = 3.\overline{63}$ .

Smaller Bound: $2.\overline{81}$

Step 7

Every real root on $p (x) = 2.75 x^{4} - 3 x^{2} + 5 x + 2$ lies between $- 2.\overline{81}$ and $2.\overline{81}$ .

$- 2.\overline{81}$ and $2.\overline{81}$