Polynomial functions

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Polynomials are some of our favorite functions.

The functions you are most familiar with are probably polynomial functions.

1 What are polynomial functions?

A polynomial function in the variable $x$ is a function which can be written in the form

\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0 \]

where the $a_i$’s are all constants (called the coefficients) and $n$ is a whole number (called the degree when $n\ne 0$). The domain of a polynomial function is $(-\infty ,\infty )$.

This means the function $f(x)=x$ is a polynomial of degree $1$, with $a_1 = 1$ and $a_0 = 0$. The function $g(x) = 2$ is a polynomial with $n=0$ and $a_0 = 2$. Similarly, any constant function is a polynomial function. However, $h(x) = \sqrt [3]{x} = x^{1/3}$ is not a polynomial since the exponent $1/3$ is not a whole number.

Which of the following are polynomial functions?

$f(x) = 0$ $f(x) = -9$ $f(x) = 3x+1$ $f(x) = x^{1/2}-x +8$ $f(x) = -4x^{-3}+5x^{-1}+7-18x^2$ $f(x) = (x+1)(x-1)+e^x - e^x $ $f(x) = \frac {x^2 - 3x + 2}{x-2}$ $f(x) = x^7-32x^6-\pi x^3+45/84$

The phrase above “in the variable $x$” can actually change.

\[ y^2-4y +1 \]

is a polynomial in $y$, and

\[ \sin ^2(x) + \sin (x) -3 \]

is a polynomial in $\sin (x)$.

2 What can the graphs look like?

Fun fact:

The Fundamental Theorem of Algebra Every polynomial of the form

\[ a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \]

where the $a_i$’s are real (or even complex!) numbers and $a_n \ne 0$ has exactly $n$ (possibly repeated) complex roots.

Remember, a root is where a polynomial is zero. The theorem above is a deep fact of mathematics. The great mathematician Gauss proved the theorem in 1799 for his doctoral thesis.

The upshot as far as we are concerned is that when we plot a polynomial of degree $n$, its graph will cross the $x$-axis at most $n$ times. Each crossing corresponds to a real root of that polynomial. (Complex roots do not give crossings!)

Sketch a rough graph of the function $f(x) = 2(x-1)^2(x+2)$.

We are given $f$ in factored form, so we recognize that $x=1$ is a zero of multiplicity $2$, and $x=-2$ is a zero of multiplicity $1$. That means our graph will have $x$-intercepts at those locations.

At these $x$-intercepts, will the graph pass through the $x$-axis like a line does, or will it touch the axis and turn back around, like a parabola does at it’s vertex? To answer this, look at a sign-chart for $f$.

[Picture]

Notice that around $x=-2$, only the $x+2$ factor changes signs. That means $f(x)$ will change signs, so the graph will pass through the $x$-axis at $x=-2$. At $x=1$, however, the $(x-1)^2$ factor does NOT change signs (due to the even exponent). That means $f(x)$ will turn around at that $x$-intercept. We also notice that for $x$ in the interval $(-\infty , -2)$ the function $f$ is negative and for $x$ in $(1,\infty )$ the function $f$ is positive. (Since $f$ is an odd-degree polynomial, its ‘ends’ go in opposite directions.) Putting this together, we have the following graph.

[Picture]

Here we see the the graphs of four polynomial functions.

For each of the curves, determine if the polynomial has even or odd degree, and if the leading coefficient (the one next to the highest power of $x$) of the polynomial is positive or negative.

Curve $A$ is defined by an evenodd degree polynomial with a positivenegative leading term.
Curve $B$ is defined by an evenodd degree polynomial with a positivenegative leading term.
Curve $C$ is defined by an evenodd degree polynomial with a positivenegative leading term.
Curve $D$ is defined by an evenodd degree polynomial with a positivenegative leading term.