Find the Bounds of the Zeros x^5-2x^4-4x^3-8x^2+x+2

$x^{5} - 2 x^{4} - 4 x^{3} - 8 x^{2} + x + 2$

Step 1

Write $x^{5} - 2 x^{4} - 4 x^{3} - 8 x^{2} + x + 2$ as a function.

$f (x) = x^{5} - 2 x^{4} - 4 x^{3} - 8 x^{2} + x + 2$

Step 2

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

Largest Degree: $5$

Leading Coefficient: $1$

Step 3

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

$- 2, - 4, - 8, 1, 2$

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 $1$ .

Arrange the terms in ascending order.

$b_{1} = | 1 |, | - 2 |, | 2 |, | - 4 |, | - 8 |$

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

$b_{1} = | - 8 |$

The absolute value is the distance between a number and zero. The distance between $- 8$ and $0$ is $8$ .

$b_{1} = 8 + 1$

Add $8$ and $1$ .

$b_{1} = 9$

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 $1$ .

Simplify each term.

The absolute value is the distance between a number and zero. The distance between $- 2$ and $0$ is $2$ .

$b_{2} = 2 + | - 4 | + | - 8 | + | 1 | + | 2 |$

The absolute value is the distance between a number and zero. The distance between $- 4$ and $0$ is $4$ .

$b_{2} = 2 + 4 + | - 8 | + | 1 | + | 2 |$

The absolute value is the distance between a number and zero. The distance between $- 8$ and $0$ is $8$ .

$b_{2} = 2 + 4 + 8 + | 1 | + | 2 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$b_{2} = 2 + 4 + 8 + 1 + | 2 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$b_{2} = 2 + 4 + 8 + 1 + 2$

Simplify by adding numbers.

Add $2$ and $4$ .

$b_{2} = 6 + 8 + 1 + 2$

Add $6$ and $8$ .

$b_{2} = 14 + 1 + 2$

Add $14$ and $1$ .

$b_{2} = 15 + 2$

Add $15$ and $2$ .

$b_{2} = 17$

Arrange the terms in ascending order.

$b_{2} = 1, 17$

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

$b_{2} = 17$

Step 6

Take the smaller bound option between $b_{1} = 9$ and $b_{2} = 17$ .

Smaller Bound: $9$

Step 7

Every real root on $f (x) = x^{5} - 2 x^{4} - 4 x^{3} - 8 x^{2} + x + 2$ lies between $- 9$ and $9$ .

$- 9$ and $9$

Find the Bounds of the Zeros x^5-2x^4-4x^3-8x^2+x+2

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