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 $(-\infty ,\infty )$.

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.

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.

2025-01-06 19:59:23