Problem 97 - Algebra and Trigonometry

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 because according to the Fundamental Theorem of Algebra, a polynomial of degree 20 has up to 20 roots and hence can cross the x-axis up to 20 times not once.

Step 1: Define a Polynomial Function and Degree

A degree of a polynomial is the highest exponent of the variable \(x\) in a polynomial function. Polynomial of degree 20 means that the highest exponent of \(x\) in the equation is 20.

Step 2: Interpret the Relationship of Degree and X-Intercepts

The fundamental theorem of Algebra states that a polynomial of degree \(n\) has exactly \(n\) roots, or x-intercepts, in the complex plane. This means that it can cross the x-axis up to \(n\) times. Therefore, a polynomial of degree 20 can have up to 20 x-intercepts.

Step 3: Final Explanation

Now, it should be clear that a polynomial function of degree 20 cannot cross the x-axis exactly once because it could have up to 20 different x-intercepts or roots.