Chapter 4: Q. 58 (page 178)

In Problems 57– 62, find the real zeros of f. If necessary, round to two decimal places.
$f (x) = x^{3} + 3.2 x^{2} - 7.25 x - 6.3$

Short Answer

Expert verified

The real zeros offafter rounding it two decimal places are $2, - 4.5, a n d - 0.7 .$

Step by step solution

Step 1. Given Information

The given function is $f (x) = x^{3} + 3.2 x^{2} - 7.25 x - 6.3 .$

We have to find the real zeros and round them to two decimal places.

Step 2. Finding the real zeros

The degree of the polynomial is $3$ ,thus it has three roots.

Since there are noninteger coefficients, the Rational Zeros Theorem doesn't apply.

Step 3. Determine the bounds on the zeros of f

The leading coefficient of f is $1$ with $a_{2} = 3.2, a_{1} = - 7.25, a n d a_{0} = - 6.3 .$

Step 4. Evaluate the expressions using the formula

So,

$M a x, {1, |3.2| + |- 7.25| + |- 6.3|} = M a x {1, 16.75} = 16.75 1 + M a x {|3.2|, |- 7.25|, |- 6.3|} = 1 + 7.25 = 8.25$

The smaller of two numbers, $16.75,$ is the bound.

Every real zero off liesbetween $- 16.75 a n d 16.75 .$

Step 5. Sketch the graph of the polynomial

From the graph, we conclude that fappears to have three x-intercepts: one between $0 a n d - 1,$ one between $- 4 a n d - 5,$ and one near $2 .$

Step 6. Use the Factor Theorem

$f (2) = {(2)}^{3} + 3.2 {(2)}^{2} - 7.25 (2) - 6.3 f (2) = 8 + 12.8 - 14.5 - 6.3 f (2) = 0$

Therefore, $2$ is the zero of the polynomial.

Using Zero we find that the remaining zeros are $- 4.5 a n d - 0.7,$ rounded to two decimal places.

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In Problems 57– 62, find the real zeros of f. If necessary, round to two decimal places.
$f (x) = x^{3} + 3.2 x^{2} - 7.25 x - 6.3$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the real zeros

Step 3. Determine the bounds on the zeros of f

Step 4. Evaluate the expressions using the formula

Step 5. Sketch the graph of the polynomial

Step 6. Use the Factor Theorem

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In Problems 57– 62, find the real zeros of f. If necessary, round to two decimal places.f(x)=x3+3.2x2-7.25x-6.3

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the real zeros

Step 3. Determine the bounds on the zeros of f

Step 4. Evaluate the expressions using the formula

Step 5. Sketch the graph of the polynomial

Step 6. Use the Factor Theorem

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Pure Maths

Statistics

Mechanics Maths

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Problems 57– 62, find the real zeros of f. If necessary, round to two decimal places.
$f (x) = x^{3} + 3.2 x^{2} - 7.25 x - 6.3$