Predict the behavior of polynomial functions.

p137 13–17, 19–32

Polynomials are expressions that involve addition, subtraction, multiplication and exponentiation to nonnegative integer powers. You’ve seen polynomials already with linear and quadratic functions, but this chapter will look at polynomials in general, rather than specific ones.

Classifying Polynomials

First thing is to lay the ground work for how we describe polynomials. First up is the standard form of a polynomial, which is when the polynomial is written so that each term is in descending order of degree, or exponent. Terms are what is being added (or subtracted).

\[-4x + 9 + 2x^3 \quad \rightarrow \quad 2x^3 - 4x + 9\]

The polygonal above has three terms and was rearranged so that the term with the highest degree was first, followed by the second highest, and so on.

Speaking of terms, the words monomial, binomial, and trinomial are used often and refer to polynomials with 1, 2, or 3 terms respectively.

The degree of a polynomial is just the greatest degree in the polynomial. In the one above, the degree of the polynomial is 3. There are also names for each degree, so while you will hear “degree 3 polynomial”, you’ll also hear it called a “cubic”.

Degree Name Example
0 Constant $4$
1 Linear $2x+3$
2 Quadratic $3x^2+x-9$
3 Cubic $x^3-6x^2+3$
4 Quartic $5x^4-2x+4$

Note that you don’t need a term for each successive degree term. The polynomial $2x^2$ is just as much a quadratic as $2x^2 + 5x -4$.

The last bit of language to cover is the leading term and leading coefficient. The leading term of a polynomial is the term with the highest degree. Coefficients are the numerical factors of a term, like the 3 in $3x$. The leading coefficient, it follows, is the one in the leading term. $6x^5 - 5$ has a leading term of $6x^5$ with a leading coefficient of 6.

End Behavior of a Polynomial

It’s helpful to get a feel for what the graphs of polynomials look like, without having the graph them. For now, we’ll focus on the extreme ends of them and leave the middle for another time. Specifically, we’ll look at whether or not the graph goes up (positive) or down (negative) at the ends.

Before getting into it, let’s look at why the leading term and coefficient are so important. Take the polynomial $2x^6-6x^5-7$ and plug in one billion, or some other really large number. So, one billion raised it to the sixth power and multiplied it by 2. You get a ridiculously large positive number.

Now the next term. Take a billion, raise it to the fifth power and multiply it by 6. There’s no way that number will have any impact on what we got from our leading term, even if we are subtracting it.

This is why there is an emphasis on the leading term and coefficient. That one term overpowers every other term when looking at the extreme ends of the function: the really large positive and negative numbers.

Now, on to getting end behavior from the leading term. We’ll use the fact that even exponents always produce positives to our advantage.

  1. Is the degree even? Both ends will point in the same direction: up.
  2. If the degree is odd, then the left side will be negative and the right side positive, or down to up.
  3. If the leading coefficient is negative, flip everything. Even degrees will point down, and odds will go from up to down.

Graphing Polynomials

I suggest a heavy dose of Desmos, or some other graphing utility. Whenever you run into a, exercise involving graphing, do your best to make a sketch on your own, then verify with technology. The book mentions turning points, but these are tough to find exactly without either a graphing calculator or calculus.

Some old knowledge will still apply here, and to any function. For example, the $y$-intercept will be when $x$ is zero, and vice versa. If you need the average rate of change, use the two given points to calculate slope.