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Chapter 3: Problem 97

Explain why a polynomial function of degree 20 cannot cross the \(x\) -axis exactly once.

Short Answer

Expert verified
A polynomial function of degree 20 cannot cross the x-axis exactly once due to the Fundamental Theorem of Algebra, which states that a polynomial function of degree n can cross the x-axis up to n times. For a polynomial function to cross the x-axis exactly once, it must be of degree 1.

Step by step solution

01

Understanding the Polynomial Functions

In mathematics, a polynomial function of degree n has the potential to intersect the x-axis up to n times. The Fundamental Theorem of Algebra supports this, stating that an nth degree polynomial has n roots. However, these roots are in the complex plane, and are not necessarily distinct.
02

Relating to the problem

In our problem, we have a 20th degree polynomial. This indicates that the polynomial function can have up to 20 real roots, i.e., it can intersect the x-axis up to 20 times. If one of these roots has multiplicity more than 1, it might touch the axis without crossing it at that point, but all in all, it can touch or cross the x-axis up to 20 times.
03

Determining the Possibility

So, when it comes to the possibility of the polynomial function crossing the x-axis exactly once, it isn't supported by the Fundamental Theorem of Algebra or the nature of polynomial functions. In order for a polynomial function to cross the x-axis exactly once, it must be of degree 1.

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Most popular questions from this chapter

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations. $$g(x)=\frac{2 x-9}{x-4}$$

Use a graphing utility to graph \(y-\frac{1}{x}, y-\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same viewing rectangle. For odd values of \(n,\) how does changing \(n\) affect the graph of \(y-\frac{1}{x^{n}} ?\)

What is a polynomial incquality?

Exercises \(110-112\) will help you prepare for the material covered in the next section. If \(S-\frac{k A}{P}\), find the value of \(k\) using \(A-60,000, P-40\) and \(S-12.000\)

a. Use a graphing utility to graph \(y-2 x^{2}-82 x+720\) in a standard viewing rectangle. What do you observe? b. Find the coordinates of the vertex for the given quadratic function. c. The answer to part (b) is \((20.5,-120.5) .\) Because the leading coefficient, \(2,\) of the given function is positive, the vertex is a minimum point on the graph. Use this fact to help find a viewing rectangle that will give a relatively complete picture of the parabola. With an axis of symmetry at \(x=20.5,\) the setting for \(x\) should extend past this, so try \(\mathrm{Xmin}=0\) and \(\mathrm{Xmax}=30 .\) The setting for \(y\) should include (and probably go below) the \(y\) -coordinate of the graph's minimum y-value, so try Ymin \(=-130\). Experiment with Ymax until your utility shows the parabola's major features. d. In general, explain how knowing the coordinates of a parabola's vertex can help determine a reasonable viewing rectangle on a graphing utility for obtaining a complete picture of the parabola.
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