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Chapter 2: Problem 3

A polynomial function of degree \(n\) has at most _____ real zeros and at most _____ turning points.

Short Answer

Expert verified
A polynomial function of degree \(n\) has at most \(n\) real zeros and at most \(n-1\) turning points.

Step by step solution

01

Understanding Polynomials

A polynomial is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s). The degree of the polynomial is the highest power among those terms.
02

Real Zeros of a Polynomial

A real zero of a polynomial is the x-value where the polynomial equals zero. It is also referred to as the root or solution of the polynomial function. According to the Fundamental Theorem of Algebra, a polynomial of degree \(n\) has exactly \(n\) complex roots. However, if we specifically talk about real zeros then it can have at most \(n\) real zeros.
03

Turning Points of a Polynomial

A turning point of a polynomial is the point where the polynomial changes from increasing to decreasing (rising to falling) or from decreasing to increasing (falling to rising). The maximum number of turning points a polynomial function can have is \(n-1\), where \(n\) is the degree of the polynomial.

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