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Chapter 4: Q. 68 (page 195)

In Problems 65–68, construct a polynomial function that might have the given graph. (More than one answer may be possible.)

Short Answer

Expert verified

The possible function is $f (x) = - \frac{1}{2} (x + 1) (x - 1) (x - 2)$

Step by step solution

01

Step 1. Given information

The graph is

02

Step 2. To find the possible function from graph.

In the given graph we can see that there are 3 zeros because the graph is crossing the x-axis at

$x = - 1, 1, 2$ , so these zeros have odd multiplicity

So the factors of the function are:

$(x + 1), (x - 1), (x - 2)$

the function is

role="math" localid="1646322635488" $f (x) = a (x + 1) (x - 1) (x - 2)$

03

Step 3. To find value of a

Since, $f (0) = - 1$

hence

$- 1 = a (0 + 1) (0 - 1) (0 - 2) - 1 = a (1) (- 1) (- 2) - 1 = 2 a a = - \frac{1}{2}$

so the function is

$f (x) = - \frac{1}{2} (x + 1) (x - 1) (x - 2)$

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Most popular questions from this chapter

True or False : The graph of $f (x) = \frac{x}{x - 3}$ is above the x-axis for $x < 0$ or $x > 3$ , so the solution set of the inequality $\frac{x}{x - 3} \geq 0$ is ${x : x \leq 0 o r x \geq 3}$ ..

In Problems 63–72, find the real solutions of each equation.

$x^{4} + 4 x^{3} + 2 x^{2} - x + 6 = 0$

Find the domain of the rational function.

$R (x) = \frac{x}{x^{4} - 1}$

Solve the inequality algebraically

$x^{3} - 2 x^{2} - 3 x > 0$

In Problems 21–32, determine the maximum number of real zeros that each polynomial function may have. Then list the potential rational zeros of each polynomial function. Do not attempt to find the zeros.

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

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