Find the Bounds of the Zeros f(x)=x^3-10x^2+27x-12

$f (x) = x^{3} - 10 x^{2} + 27 x - 12$

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: $1$

Step 2

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

$- 10, 27, - 12$

Step 3

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.

$b_{1} = | - 10 |, | - 12 |, | 27 |$

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

$b_{1} = | 27 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $27$ is $27$ .

$b_{1} = 27 + 1$

Add $27$ and $1$ .

$b_{1} = 28$

Step 4

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 $- 10$ and $0$ is $10$ .

$b_{2} = 10 + | 27 | + | - 12 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $27$ is $27$ .

$b_{2} = 10 + 27 + | - 12 |$

The absolute value is the distance between a number and zero. The distance between $- 12$ and $0$ is $12$ .

$b_{2} = 10 + 27 + 12$

Simplify by adding numbers.

Add $10$ and $27$ .

$b_{2} = 37 + 12$

Add $37$ and $12$ .

$b_{2} = 49$

Arrange the terms in ascending order.

$b_{2} = 1, 49$

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

$b_{2} = 49$

Step 5

Take the smaller bound option between $b_{1} = 28$ and $b_{2} = 49$ .

Smaller Bound: $28$

Step 6

Every real root on $f (x) = x^{3} - 10 x^{2} + 27 x - 12$ lies between $- 28$ and $28$ .

$- 28$ and $28$

Find the Bounds of the Zeros f(x)=x^3-10x^2+27x-12

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