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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

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(79-82 .\) Then determine the number of real zeros and the number of imaginary zeros for each function. $$f(x)=3 x^{5}-2 x^{4}+6 x^{3}-4 x^{2}-24 x+16$$

Write an equation in standard form of the parabola that has the same shape as the graph of \(f(x)=3 x^{2}\) or \(g(x)=-3 x^{2},\) but with the given maximum or minimum. Maximum \(=-7\) at \(x=5\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with a fourth-degree polynomial function with integer coefficients and zeros at 1 and \(3+\sqrt{5} .\) I'm certain that \(3+\sqrt{2}\) cannot also be a zero of this function.

Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+4}{x}>0 $$

Find the slant asymptote of the graph of each rational function and \(\mathbf{b}\). Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$ f(x)=\frac{x^{2}+4}{x} $$
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