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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 because, according to the Fundamental Theorem of Algebra, this polynomial should have exactly 20 roots. This means it should cross the \(x\)-axis at least 20 times, considering each root's multiplicity.

Step by step solution

01

Understanding the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree \(n\) will have exactly \(n\) roots, considering multiplicity (i.e., considering each root's multiplicity). This means that a polynomial function crosses the \(x\)-axis at each of its real roots. Real roots are the roots for which the function value is zero.
02

Applying the Theorem

Applying this theorem to a polynomial of degree 20, it implies that it should have exactly 20 roots. This means that the polynomial crosses the \(x\)-axis at each root.
03

Concluding the Reasoning

Therefore, it is impossible for a polynomial function of degree 20 to cross the \(x\)-axis exactly once. It must cross the \(x\)-axis at least 20 times, considering each root's multiplicity.

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