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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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Most popular questions from this chapter

The equations in Exercises \(72-75\) have real roots that are rational. Use the Rational Zero Theorem to list all possible rational roots. Then graph the polynomial function in the given viewing rectangle to determine which possible rational roots are actual roots of the equation. $$2 x^{3}-15 x^{2}+22 x+15=0 ;[-1,6,1] \text { by }[-50,50,10]$$

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