A polynomial is a function of the form
$$P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0.$$ The number $n$ is called the degree of the polynomial $P(x)$. The term $a_nx^n$ is called the leading term and $a_n$ is called the leading coefficient. The number $a_0$ is called the constant term. $P(x)$ is called linear if $n=1$, quadratic if $n=2$, cubic if $n=3$, quartic if $n=4$, quintic if $n=5$, sextic if $n=6$, septic if $n=7$, and so on so forth. You don’t really have to worry about memorizing these jargons. Names are less important. However you need to remember at least what the degree is and what the leading coefficient is.
The Leading Term Test
There is a pattern for the long term behavior of a polynomial, i.e. the behavior of a polynomial when $x\to\infty$ or $x\to -\infty$. The behavior can be characterized as follows.
- $n=\mbox{even}$ and $a_n>0$:
Example. $f(x)=3x^4-2x^3+3$
- $n=\mbox{even}$ and $a_n<0$:
Example. $f(x)=-x^6+x^5-4x^3$
- $n=\mbox{odd}$ and $a_n>0$:
Example. $f(x)=x^5+\frac{1}{4}x+1$
- $n=\mbox{odd}$ and $a_n<0$:
Example. $f(x)=-5x^3-x^2+4x+2$
Finding zeros of a polynomial $P(x)$
By factoring, solve the equation $P(x)=0$. The solutions are the zeros of $P(x)$.
Example. Find the zeros of $P(x)=x^3+2x^2-5x-6$.
Solution. \begin{align*}
P(x)&=x^3+2x^2-5x-6\\
&=(x^3+x^2)+(x^2-5x-6)\ \mbox{(grouping)}\\
&=x^2(x+1)+(x-6)(x+1)\\
&=(x+1)(x^2+x-6)\\
&=(x+1)(x+3)(x-2).
\end{align*}
Hence, $P(x)$ has zeros $x=-3,-1,2$.
How do we determine whether $x=a$ is a zero of a polynomial $P(x)$?
To only check whether $x=a$ is a zero of $P(x)$, you don’t really have to factor $P(x)$. This is what you need to know. If $P(a)=0$, then $x=a$ is a zero of the polynomial $P(x)$.
Example. Consider $P(x)=x^3+x^2-17x+15$. Determine whether each of numbers 2 and $-5$ is a zero of $P(x)$.
Solution. $P(2)=(2)^3+(2)^2-17(2)+15=-7$, so $x=2$ is not a zero. $P(-5)=(-5)^3+(-5)^2-17(-5)+15=0$, so $x=-5$ is a zero of $P(x)$.
Even and Odd Multiplicity
Even and odd multiplicity is an important property for sketching the graph of a polynomial function. Suppose that $k$ is the largest integer such that $(x-c)^k$ is a factor of $P(x)$. The number $k$ is called the multiplicity of the factor $x-c$.
- If $k$ is odd, the graph of $P(x)$ crosses the $x$-axis at $(c,0)$.
- If $k$ is even, then the graph of $P(x)$ is tangent to the $x$-axis, i.e. touches the $x$-axis without crossing at $(c,0)$.
Example. Consider $f(x)=x^2(x+3)^2(x-4)(x+1)^4$. The factors $x$ and $x+3$ have multiplicity 2 and the factor $x+1$ has multiplicity 4. Hence the graph of $f(x)$ touches the $x$-axis without crossing at $x=0$, $x=-3$ and $x=-1$. The factor $x-4$ has multiplicity 1, so the graph crosses the $x$-axis at $x=4$. This is also shown in the following figure.
