Chapter 4: Q. 22 (page 245)

Find bounds to the real zeros of each polynomial function. Obtain a complete graph of f using a graphing utility.
$f (x) = 2 x^{3} - 7 x^{2} - 10 x + 35$

Short Answer

Expert verified

Every zero of polynomial lies between $- \frac{37}{2} a n d \frac{37}{2}$ and the graph is

Step by step solution

Step 1. Given Information

The given polynomial is $f (x) = 2 x^{3} - 7 x^{2} - 10 x + 35$

Step 2. Explanation

The leading coefficient of the polynomial is 2. So, first we will write the function such that the leading coefficient is 1.

$f (x) = 2 (x^{3} - \frac{7}{2} x^{2} - \frac{10}{2} x + \frac{35}{2}) = 2 (x^{3} - \frac{7}{2} x^{2} - 5 x + \frac{35}{2})$

Now, leading coefficient is 1 with other coefficients as $a_{2} = - \frac{7}{2}, a_{1} = - 5, a_{0} = \frac{35}{2}$

On applyiing the Max formula, we will find the bounds of the given polynomial $M a x \{1, | a_{0} | + | a_{1} | + . . . . + | a_{n - 1} |\} = M a x \{1, | \frac{35}{2} | + | - 5 | + | - \frac{7}{2} |\} = M a x {1, \frac{35 + 10 + 7}{2}} = M a x {1, \frac{52}{2}} = \frac{52}{2} 1 + M a x \{| a_{0} |, | a_{1} |, . . . . | a_{n - 1} |\} = 1 + M a x {| \frac{35}{2} |, | - 5 |, | \frac{7}{2} |} = 1 + M a x {\frac{35}{2}, 5, \frac{7}{2}} = 1 + \frac{35}{2} = \frac{37}{2}$

The smaller of the two numbers which means $\frac{37}{2}$ is the bound of the polynomial.

So, every polynomial exist between $- \frac{37}{2} a n d \frac{37}{2}$

Step 3. Graph

The graph of the given polynomial is

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Find bounds to the real zeros of each polynomial function. Obtain a complete graph of f using a graphing utility.
$f (x) = 2 x^{3} - 7 x^{2} - 10 x + 35$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

Step 3. Graph

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Find bounds to the real zeros of each polynomial function. Obtain a complete graph of f using a graphing utility.f(x)=2x3-7x2-10x+35

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

Step 3. Graph

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

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Logic and Functions

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Study anywhere. Anytime. Across all devices.

Find bounds to the real zeros of each polynomial function. Obtain a complete graph of f using a graphing utility.
$f (x) = 2 x^{3} - 7 x^{2} - 10 x + 35$