Find the Bounds of the Zeros f(x)=x^3-3x^2-25x+75

$f (x) = x^{3} - 3 x^{2} - 25 x + 75$

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

Largest Degree: $3$

Leading Coefficient: $1$

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

$- 3, - 25, 75$

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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Arrange the terms in ascending order.

$b_{1} = | - 3 |, | - 25 |, | 75 |$

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

$b_{1} = | 75 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $75$ is $75$ .

$b_{1} = 75 + 1$

Add $75$ and $1$ .

$b_{1} = 76$

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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Simplify each term.

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The absolute value is the distance between a number and zero. The distance between $- 3$ and $0$ is $3$ .

$b_{2} = 3 + | - 25 | + | 75 |$

The absolute value is the distance between a number and zero. The distance between $- 25$ and $0$ is $25$ .

$b_{2} = 3 + 25 + | 75 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $75$ is $75$ .

$b_{2} = 3 + 25 + 75$

Simplify by adding numbers.

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Add $3$ and $25$ .

$b_{2} = 28 + 75$

Add $28$ and $75$ .

$b_{2} = 103$

Arrange the terms in ascending order.

$b_{2} = 1, 103$

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

$b_{2} = 103$

Take the smaller bound option between $b_{1} = 76$ and $b_{2} = 103$ .

Smaller Bound: $76$

Every real root on $f (x) = x^{3} - 3 x^{2} - 25 x + 75$ lies between $- 76$ and $76$ .

$- 76$ and $76$

Find the Bounds of the Zeros f(x)=x^3-3x^2-25x+75

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