Polynomial Functions

These notes cover polynomial functions.

Polynomial Equations

We assume familiarity with the follow propositions and theorems.

${\forall a,m,n \in \reals:}$
${(a^m)(a^n) = a^{m+n}}$	${\dfrac{a^m}{a^n} = a^{m-n}}$
${(a^m)^n = a^{m \by n}}$	${a^0 = 1}$
${a^{-n}=\dfrac{1}{n}}$	${(ab)^n = (a^n)(b^n)}$
${\ar{\dfrac{a}{b}}=\dfrac{a^n}{b^n}}$

Linear Functions

definition. A linear function is a function of the form ${f(x)=mx+b,}$ where ${m,x,b \in \reals}$ and ${m}$ and ${b}$ are constants. We call ${b}$ the initial value of ${f,}$ and ${m}$ the slope of ${f.}$

Below are a few examples of linear functions.

Deriving Linear Functions

Linear functions are the simplest to derive (setting aside constant functions). The following formula computes the slope ${m}$ for two given arguments ${x_1, x_2 \in \dom{f}.}$

m = \dfrac{f(x_2) - f(x_1)}{x_2 - x_1}.

Quadratic Functions

Quadratic functions are functions whose procedures are quadratic expressions.

quadratic function. A quadratic function is a function of the form ${f(x)=ax^2+bx+c,}$ where ${a,b,c \in \reals}$ are constant coefficients with ${a \neq 0.}$ We call the term ${ax^2}$ the quadratic term, the term ${bx}$ the linear term, and the term ${c}$ the constant term.

The graphs below all correspond to quadratic functions.

Cubic Functions

Cubic functions are functions whose procedures are cubic expressions.

cubic function. A cubic function is a function of the form ${f(x)=ax^3 + bx^2 + cx + d,}$ where ${a,b,c,d}$ are constant coefficients with ${a \neq 0.}$ We call the term ${ax^3}$ the cubic term, the term ${bx^2}$ the quadratic term, the term ${cx}$ the linear term, and the term ${d}$ the constant term.

example. Some examples of cubic functions:

Power Functions

The square and cubic functions are types of power functions.

power function. A power function is a function of the form ${f(x)=cx^n,}$ where ${c}$ is a real constant, and ${n}$ is a natural number called the power of the base argument ${x.}$

Deriving Power Functions

Let's now consider the derivative of ${f(x) = x^n,}$ where ${n \in \uint^{+}.}$ In other words, we want to compute:

\dfrac{\d}{\d x} x^n

Consider the simplest case, where ${n = 1.}$ If ${n = 1,}$ then ${f(x) = x^n = x^1 = x.}$ The graph of ${f(x) = x}$ is a straight vertical line, which has a slope of ${1.}$ Thus, we have:

theorem. Given the function ${f(x) = x,}$ it follows that
$\dfrac{\d}{\d x}~x = 1.$

But about the more general case, ${x^n?}$ For this case, we return to our expression, ${\frac{\Delta f}{\Delta x}:}$

\dfrac{(x + \Delta x)^n - x^n}{\Delta x}

Notice that we wrote ${x}$ instead of ${x_0.}$ We did so because ${x_0}$ isn't meaninful in this particular computation. ${x}$ can be any fixed value of ${x,}$ and we aren't so concerned about its value. However, we state explicitly that ${x}$ is a fixed value, and ${\Delta x}$ is a moving value. To manipulate our expression, we must understand what the ${n^{\text{\scriptsize{th}}}}$ power of a sum is. To do so, we use a theorem from discrete mathematics, called the binomial theorem. The expression ${(x + \Delta x)^n}$ can be rewritten as:

(x+\Delta x)_1 \cdot (x+\Delta x)_2 \cdot \ldots \cdot (x+\Delta x)_n

If we expand this multiplication, the first term in the product is ${x^n.}$ Next, we have a second term of the form ${x^{n-1}\Delta x.}$ How many times does this term occur? ${n}$ times, since we are multiplying ${(x + \Delta x)}$ by itself ${n}$ times. Thus, we have:

x^n + nx^{n - 1}\Delta x

With just these terms, we have enough to continue computing the derivative, per the binomial theorem. According to the binomial theorem:

(x + \Delta x)^n = \sum\limits_{k = 0}^{n} \dbinom{n}{k}x^{n - k}(\Delta x)^k

From this proposition, when ${k \geq 2,}$ we will always obtain terms containing ${\Delta x}$ in powers of 2, 3, 4, 5, and so on. Because of this implication, we can write the sum as:

(x + \Delta x)^n = x^n + nx^{n - 1}\Delta x + O((\Delta x)^2)

The term ${O(\Delta x)^2}$ encapsulates what we call "junk terms," at least in the context of continuous mathematics, and more specifically, in the context of ${\Delta x \mapsto 0}$ . Just so we aren't hiding the ball, the term ${O(\Delta x)^2}$ represents:

(\Delta x)^2 \sum\limits_{k = 2}^{n} \binom{n}{k}x^{n - k} = (\Delta x)^2 \left( \binom{n}{2}x^{n-2} + \binom{n}{3}x^{n-3} \Delta x + \ldots - (\Delta x)^{n-2} \right)

Importantly, as ${\Delta x \to 0,}$ we have the equation:

\lim\limits_{\Delta x \to 0} \dfrac{o(\Delta x)}{\Delta x} = 0

Let's get back to our manipulation. With our application of the binomial theorem, we can rewrite our difference quotient as:

\begin{aligned} \dfrac{\Delta f}{\Delta x} &= \dfrac{1}{\Delta x}((x + \Delta x)^n - x^n) \\ &= \dfrac{1}{\Delta x}(x^n + nx^{n-1}\Delta x + O((\Delta x)^2) - x^n) \\ &= nx^{n-1} + \dfrac{O((\Delta x)^2)}{\Delta x} \end{aligned}

Now all we do is apply the limit (recall what we concluded from the binomial theorem earlier):

\begin{aligned} \lim\limits_{\Delta x \to 0} \dfrac{\Delta f}{\Delta x} &= \lim\limits_{\Delta x \to 0} \left(nx^{n-1} + \dfrac{O((\Delta x)^2)}{\Delta x}\right) \\ &= nx^{n-1} + 0 \\ &= nx^{n-1} \end{aligned}

Accordingly, we have the following theorem:

power rule. Given the function ${f(x)=x^n}$ where ${n \in \pint,}$ it follows that
$\dfrac{\d}{\d x} x^n = nx^{n - 1}.$

We can now compute the derivatives of various power functions:

\begin{aligned} \frac{\d}{\d x} x^2 &= 2x \\[1em] \frac{\d}{\d x} x^3 &= 3x^2 \\[1em] \frac{\d}{\d x} x^4 &= 4x^3 \\[1em] \frac{\d}{\d x} x^5 &= 5x^4 \end{aligned}

Polynomial Functions

Both quadratic and cubic functions are the simplest subtypes of polynomial functions — functions whose procedures are polynomials.

polynomial function. A polynomial function is a function of the form
$f(x)=a_nx^n + a_{n-1}x^{n-1}+\ldots+a_2x^{2} + a_1x + a_0,$
where each ${a_i}$ is a constant coeffient, and each ${a_ix_i^{i}}$ is a term, for all ${n,i \in \nat.}$