Polynomial Function

Polynomials Jargon. The term "polynomial" has multiple uses: (1) it is used to describe polynomial expressions, (2) it is used to describe polynomial functions, and, (3) despite the Greek prefix "poly" that means "many," it is also used to refer to solitary power functions like y=–3x7. This condensation of terminology is not unusual in mathematics: recall, for example, the various uses of "–": (1) to signify a negative number, (2) to represent subtraction between two numbers, and (3) to signify multiplication by –1, as in x = (–1)x.

Select a "Convenient" Test Value. For various purposes, when analyzing a function, it is necessary to select "test values" in certain intervals. You are permitted to select a value in each such interval that makes your calculations as easy as possible. Typically, 0 or an integer close to 0, if there is one in the given interval, is the most convenient.

Continuity and "Smoothness". Polynomial functions are continuous everywhere, and very smooth—no corners, breaks, or holes.

Maximum Number of x-Intercepts. The maximum number of x-intercepts equals the degree of the polynomial. This follows from the Fundamental Theorem of Algebra, which states that a polynomial of degree n can be expressed as a product of n linear factors of the form (ax+b), where a and b need not be real-valued. Such a linear factor gives rise to a zero, –b/a, which is also an x-intercept if it is real-valued.

Minimum Number of x-Intercepts. The minimum number of x-intercepts depends on whether the degree, n > 0, is odd or even. If n is odd, there is at least one x-intercept but if n is even, there may not be any. This follows from the End Behaviors of a polynomial, which are described below.

Maximum Number of Turning Points. The maximum number of turning points is one less than the degree of the polynomial. So, if the degree is n, the maximum number of turning points is n–1.

Minimum Number of Turning Points. The minimum number of turning points depends on whether the degree, n > 0, is odd or even. If n is even, there is at least one turning point but if n is odd, there may not be any.

End Behaviors. For polynomials of degree greater than 0, there are only 4 possible end behaviors. These behaviors are shown in the following table and illustrated below with the graphs of x2, –x2, x3, and –x3

 Degree Leading Coefficient Even Odd + ↑↑(Up left; Up right) ↓↑(Down left; Up right)  ↓↓(Down left; Down right) ↑↓(Up left; Down right)

Graphing an Elementary Polynomial in Factored Form:
Example of a Cubic: y=(x1)(x+2)2

First Approach. Plot the y-intercept (set x=0 and solve for y) and plot the x-intercept (set y=0 and solve for x). Here, the y-intercept is y=(01)(0+2)2 = 4 and the x-intercepts are determined by solving for x: 0=(x–1)(x+2)2. Setting each factor to 0, x–1=0 and (x+2)2=0, we get x=1 and x=2. You can now plot the intercepts together with the end behaviors of a cubic with positive leading coefficient (down on the left, up on the right):

You may see how to complete the curve at this point, by smoothly connecting the curves and points.

Second Approach. As above, find the x- and y-intercepts. The x-intercepts are the only places where the y-value may, but need not, change sign. So, with two x-intercepts, 2 and 1, the x-axis is split into three intervals: (∞,2), (2,1), and (1,∞) where the function is either always positive (the graph is in the upper half-plane) or always negative (the graph is in the lower half-plane). Selecting convenient "test values" on each interval, say x=3, x=0, and x=2, and substituting into the formula for y, we get

 Test value y=(x–1)(x+2)2 Sign of y –3 –4 – 0 –4 – 2 16 +

Using the signs in this table, we can plot the intercepts and then sketch the graph near the x-intercepts in the correct half-plane on each side (a sign of "+" is sketched in the top half-plane, a sign of "" is sketched in the bottom half-plane):

The final step is to smoothly connect the points and extend the graph at the ends. As a check, note that because the multiplicity of the zero x= 2 is even (the factor (x+2) appears two times in the factored form) , the graph touches but does not cross the x-axis and because the multiplicity of the zero x=1 is odd (the factor (x–1) appears one time in the factored form), the graph crosses the x-axis.

Intervals of Increase and Decrease. Should the endpoints of an interval of increase or decrease be included in the interval? Here, we are referring to an interval property and not a point property, so the endpoints are not essential to reflect the underlying geometry. In fact, the property could easily be restricted to open intervals to avoid this issue. Textbook authors vary in their usage and you will want to follow your textbook author and your teacher's usage on a test. However, working within the standard definitions (f is increasing means: a < b implies f (a) < f (b); f is decreasing means: a < b implies f (a) > f (b)), the endpoints could be included if the function is defined at the endpoints. For polynomials, this is always the case.

Intervals of Concavity. Should the endpoints of an interval where a function is concave up or concave down be included in the interval? In general, concavity is an interval property and not a point property, so the endpoints are not essential to reflect the underlying geometry. In fact, the property could easily be restricted to open intervals to avoid this issue. Textbook authors vary in their usage and you will want to follow your textbook author and your teacher's usage on a test. However, working within standard definitions (f is concave up on the interval I means: f' is increasing on I; f is concave down on the interval I means: f' is decreasing on I), the endpoints could be included if the derivative, f', is defined at the endpoints. For polynomials, this is always the case. For nonpolynomials, it needs to be checked.

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