Lesson 8

What do you notice? What do you wonder?

1. For your assigned polynomial, complete the column for the different values of \(x\). Discuss with your group what you notice.

   \(x\)	\(y = x^2 + 1\)	\(y = x^3 + 1\)	\(y = x^4 + 1\)	\(y = x^5 + 1\)
   -1000
   -100
   -10
   -1
   1
   10
   100
   1000

2. Sketch what you think the end behavior of your polynomial looks like, then check your work using graphing technology.

Consider the polynomial \(y=2x^5 - 5x^4 - 30x^3 + 5x^2 + 88x + 60\).

1. Identify the degree of the polynomial.
2. Which of the 6 terms, \(2x^5\), \(5x^4\), \(30x^3\), \(5x^2\), \(88x\), or \(60\), is greatest when:
   1. \(x=0\)
   2. \(x=1\)
   3. \(x=3\)
   4. \(x=5\)
3. Describe the end behavior of the polynomial.

We know that if the expression for a polynomial function \(f\) written in factored form has the factor \((x-a)\), then \(a\) is a zero of \(f\) (that is,\(f(a)=0\)) and the point \((a, 0)\) is on the graph of the function. But what about other values of \(x\)? In particular, as we consider values of \(x\) that get larger and larger in either the negative or positive direction, what happens to the values of \(f(x)\)?

The answer to this question depends on the degree of the polynomial, because any negative real number raised to an even power results in a positive number. For example, if we graph \(y=x^2\), \(y=x^3\) and \(y=x^4\) and zoom out, we see the following:

For both \(y=x^2\) and \(y=x^4\), large positive values of \(x\) or large negative values of \(x\) each result in large positive values of \(y\). But for \(y=x^3\), large positive values of \(x\) result in large positive values of \(y\), while large negative values of \(x\) result in large negative values of \(y\).

Consider the polynomial \(P(x)=x^4-30x^3-20x^2+1000\). The leading term, \(x^4\), almost seems smaller than the other 3 terms. For certain values of \(x\), this is even true. But, for values of \(x\) far away from zero, the leading term will always have the greatest value. Can you see why?

\(x\)	\(x^4\)	\(\text-30x^3\)	\(\text-20x^2\)	\(1000\)	\(P(x)\)
-500	62,500,000,000	3,750,000,000	-5,000,000	1,000	66,245,001,000
-100	100,000,000	30,000,000	-200,000	1,000	129,801,000
-10	10,000	30,000	-2,000	1,000	39,000
0	0	0	0	1,000	1000
10	10,000	-30,000	-2,000	1,000	-21,000
100	100,000,000	-30,000,000	-200,000	1,000	69,801,000
500	62,500,000,000	-3,750,000,000	-5,000,000	1,000	58,745,001,000

The value of the leading term \(x^4\) determines the end behavior of the function, that is, how the outputs of the function change as we look at input values farther and farther from 0. In the case of \(P(x)\), as \(x\) gets larger and larger in the positive and negative directions, the output of the function gets larger and larger in the positive direction.