Problem 97 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.