Section 3.1: Polynomial Functions and Their Graphs A function P that can be written in the form P (x) = an xn +· · ·+a1 x+a0 with an , . . . , a1 , a0 being real numbers and an ̸= 0 is called a polynomial function of degree n.

The graphs of polynomial functions are continuous and smooth. That means that a function whose graph has breaks, holes, corners, or cusps would not be a polynomial function.

We have have looked at graphing basic polynomials functions via transformations.

Real Zeroes of Polynomial Functions Let P be a polynomial and c be a real number. The following are equivalent: i) x = c is a zero of P . ii) x = c is a solution of P (x) = 0. iii) (x − c) is a factor of P (x). iv) x = c is an x-intercept of the graph of P .

Recall that we looked at graphing polynomials by finding the x-intercepts and then testing test values in the intervals partitioned by the x-intercepts [to see if the graph was above or below the x-axis in each interval]. Exercise: Graph the polynomial function by finding its zeroes and then testing test values. f (x) = x4 − 4x2

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We look at yet another way to graph polynomials. It involves the end behavior of polynomials and the multiplicities of the zeroes.

End Behavior of Polynomials The end behavior of the polynomial P (x) = an xn + · · · a1 x + a0 is determined by the sign of the leading coefficient an and the degree n, as indicated below. Leading Coefficient Test • If an > 0, then f (x) → ∞ as x → ∞. That is, the graph“rises to the right”. • If an < 0, then f (x) → −∞ as x → ∞. That is, the graph “falls to the right”. Degree Test • If n is even, then f (x) → ±∞ as x → −∞ and f (x) → ±∞ as x → ∞. That is, the “right-hand and left-hand behavior of the graph will be the same”. • If n is odd, then f (x) → ±∞ as x → −∞ and f (x) → ∓∞ as x → ∞. That is, the “right-hand and left-hand behavior of the graph are opposite each other”.

Exercises: Describe the orientation of the left-hand and right-hand sides of the graph of each polynomial function. i) f (x) = 5x4 − 2x2 − 3x + 1

2 ii) g(x) = − x7 − 3x5 + 20x3 + 19 3 2

Repeated Real Zeroes Suppose P is a polynomial where (x − c)m , a factor of P and m > 1. Then x = c is a zero of multiplicity m. i) If m is odd, then the graph crosses the x-axis at x = c. ii) If m is even, then the graph touches the x-axis [but does not cross the x-axis] at x = c.

Exercises: (a) Determine the orientation of the left-hand and right-hand sides of the graph of each polynomial function, (b) find all the real zeroes of the polynomial function, (c) determine the multiplicity of each zero and describe what the graph does at each zero, and (d) sketch a generic graph of the polynomial function. i) f (x) = −x4 + x3 + 20x2

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ii) g(x) = x3 − 4x2 − 25x + 100

Fundamental Theorem of Algebra A polynomial function P of degree n has a most n real zeroes.

Relative Extrema of Polynomials A polynomial function P of degree n has at most n − 1 turning points or relative extrema points.

Additional Practice Exercises pp. 262-264 Nos. 3-17 odd, 23-35 odd, 82 4