Polynomials

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characteristics

Polynomials

Polynomial functions (over the real numbers) are functions that can be described as sums of power functions with whole number exponents.

\[ p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_3 x^3 + a_2 x^2 + a_1 x^1 + a_0 x^0 \]

where the $a_k$ are real numbers and $a_n \ne 0$.

Each of the summands is called a term.
$n$ is called the degree of the polynomial.
$a_k$ are called coefficients.
$a_n$ is called the leading coefficient.
$a_n x^n$ is called the leading term.
$a_2 x^2$ is called the quadratic term.
$a_1 x^1$ is called the linear term, often written as $a_1 x$.
$a_0 x^0$ is called the constant term, often written as $a_0$.

This is called the standard form.

When analyzing the behavior of polynomials, the characteristcis depend on the zeros of the derivative, which Calculus will show us is another polynomial. Analyzing polynomials rests on their zeros.

For analytic purposes, we would prefer a product

\[ p(x) = a (x-r_n)(x-r_{n-1}) \cdots (x-r_2)(x-r_1) \]

$a$ is called the leading coefficient.
$(x-r_k)$ are called factors.
$r_k$ are called the zeros or roots.

Polynomials are the most familiar feeling functions we have. Their formulas look like extended linear or quadratic functions, just more terms. Their graphs are smoothly flowing hills and valleys.

The graph of $y = f(x) = \frac {1}{100} x^3 + \frac {1}{25} x^2 - \frac {31}{100} x - \frac {7}{10}$

All of the characteristics and features of polynomials are nice.

Their domains include all real numbers.
They are continuous everywhere.
No discontinuities or singularities.
No asymptotes on the graph.
Their graphs are smooth - no corners or endpoints

Polynomials smoothly alternate between increasing and decreasing, which switch at global and local maximums and minimums.

As you will see later, polynomials have a limit on the number of zeros and extrema they can have. They cannot have more than the degree of the polynomial. So, what happens outside all of the wiggling? This is known as the end-behavior. All polynomials continue without bound after the zeros and extrema. Either they tend to infinity or negative infinity, which is tied to the sign of the leading coefficient and the degree.

Note: Constant functions are technically polynomials. They have a completely different behavior than what we think of with polynomials.

Factored Form

$p(x) = a (x-r_n)(x-r_{n-1}) \cdots (x-r_2)(x-r_1)$ is the factored form of a polynomial.

In an effort to be clearer, we clean up the roots, because there could be repeated roots and we would like to point this out. Therefore the reduced or simplified factored form looks like

\[ p(x) = a (x-r_n)^{e_n} (x-r_{n-1})^{e_{n-1}} \cdots (x-r_2)^{e_2} (x-r_1)^{e_1} \]

In this version, the $r_k$ are distinct roots. They are all different.
The $e_k$ are the exponents and tell us how many times a particular root repeats. The exponents are referred to as the factor’s or the root’s multiplicity.

Multiplicity

Let $p(x)$ be a polynomial with root $r$. The number of times $(x - r)$ appears in the complete factorization of $p(x)$ is called the multiplicity of $r$.

As we saw with quadratics, this factorization into linear factors often requires complex numbers, which we will encounter later in this course. We are only using real numbers now. Therefore, our factorizations will include linear factors as well as irreducible quadratic factors (quadratics nonreal/complex roots).

Running the Number Line

If we think of the real numbers as lining up to form the number line, then we could imagine ourselves running over the number line from $-\infty $ to $\infty $, or from left to right. We would hit every real number once.

For a polynomial, we would run through each distinct root once. As we ran from the left side to the right side of each root, the sign of the value of the polynomial with either change or stay the same.

At each root, $r_k$, the value of the polynomial is $0$. But on either immediate side of the root, the polynomial is either positive or negative. There are four possible combinations.

\[ \begin{array}{rcl} \text {Left of Root} & \text {Root} & \text {Right of Root} \\ positive & 0 & positive \\ negative & 0 & negative \\ positive & 0 & negative \\ negative & 0 & positive \end{array} \]

Either the sign switches or it stays the same across the root. How can we see this in the formula?

\[ f(x) = a (x-r_n)^{e_n} (x-r_{n-1})^{e_{n-1}} \cdots (x-r_2)^{e_2} (x-r_1)^{e_1} \]

If we imagine ourselves running through $r_k$ in the domain, then $(x-r_k)^{e_k}$ is the only factor that could possibly change sign. All of the other factors maintain their same sign when $x$ runs through $r_k$.

Let’s examine the factor $(x-r_k)^{e_k}$:

As $x$ runs from less than $r_k$ to greater than $r_k$, the base of this factor changes from negative to positive:

when $x<r_k$, then $x-r_k <0$.
when $x>r_k$, then $x-r_k >0$.

If the exponent, $e_k$, is even, then the sign of $x-r_k$ doesn’t matter. The sign of the whole factor $(x-r_k)^{e_k}$ will be positve regardless of the sign of the base.

However, if $e_k$ is odd then the sign of the base does matter, because a negative number raised to an odd power will remain negative.

If $e_k$ is odd, then $(x-r_k)^{e_k}$ will change sign.
If $e_k$ is even, then $(x-r_k)^{e_k}$ will not change sign.

The multiplicity or exponent tells us if the sign changes across the zero or singularitiy.

If $r_k$ is a root of odd multiplicty, then $(x-r_k)^{e_k}$ will change sign as $x$ runs through $r_k$ in the domain.
If $e_k$ is a root of even multiplicty, then $(x-r_k)^{e_k}$ will not change sign as $x$ runs through $r_k$ in the domain.

Signs

Let $p(x) = 3(x+4) (x-1)^2 (x-3)$.

$p$ has four different factors: $3$, $x+4$, $x-1$, and $x-3$. The factor $3$ is always positive, but the other three factors might change signs over their corresponding zeros, which are $-4$, $1$, and $3$.

$\blacktriangleright $ The $x+4$ factor:

$\blacktriangleright $ The $x-1$ factor:

Except, this factor actually has an even exponent, which means the factors will not change signs over $1$.

$\blacktriangleright $ The $(x-1)^2$ factor:

$\blacktriangleright $ The $x-3$ factor:

The polynomial function has three zeros: $-4$, $1$, and $3$.

In between these zeros, the values of the polynomial function has a positive or negative sign. It can’t change, since polynomilas are continuous.

The sign of the whole polynomial depends on the product of the signs of the individual factors.

For $x < -4$,
- $3 > 0$
- $x+4 < 0$
- $(x-1)^2 > 0$
- $x-3 < 0$
In the interval; $(-\infty , -4)$, the whole polynomial is $pos \cdot neg \cdot pos \cdot neg = positive$. We also know this because $p$ is a fourth degree polynomial and the leading coefficient is positive. That tell us that the end-behvior is unbounded positively in both directions.
For $-4 < x < 1$,
- $3 > 0$
- $x+4 > 0$
- $(x-1)^2 > 0$
- $x-3 < 0$
In the interval; $(-4, 1)$, the whole polynomial is $pos \cdot pos \cdot pos \cdot neg = negative$. This also agrees with the fact that $p$ was positive on $(-\infty , -4)$ and MUST change signs over $-4$, since it has an odd multiplicity.
For $1 < x < 3$,
- $3 > 0$
- $x+4 > 0$
- $(x-1)^2 > 0$
- $x-3 < 0$
In the interval; $(1, 3)$, the whole polynomial is $pos \cdot pos \cdot pos \cdot neg = negative$. This also agrees with the fact that $p$ was negative on $(-4,1)$ and MUST NOT change signs over $1$, since it has an even multiplicity.
For $3 < x$,
- $3 > 0$
- $x+4 > 0$
- $(x-1)^2 > 0$
- $x-3 > 0$
In the interval; $(3, \infty )$, the whole polynomial is $pos \cdot pos \cdot pos \cdot pos = positive$. This also agrees with the fact that $p$ was negative on $(1,3)$ and MUST change signs over $3$, since it has an odd multiplicity.

$\blacktriangleright $ The whole polynomial $3(x+4)(x-1)^2(x-3)$

All of the information about the sign of the polynomial is encoded in the zeros and their exponents.

In this example, our polynomial has a positive leading coefficient and is of degree $4$. Any fourth degree polynomial with a positive leading coefficient approaches $\infty $ as the domain approaches $-\infty $. Our polynomial is positive for domain numbers less than $-4$.

Then, this polynomial must change signs at $-4$, because $-4$ has odd multiplicity.

Then, this polynomial cannot change signs at $1$, since it has even multiplicity.

Then, this polynomial must change signs at $3$.

Finally, any fourth degree polynomial with a positive leading coefficient approaches $\infty $ as the domain approaches $\infty $.

All of this should agree with the graph of the polynomial.

Graphically

Graphically, the values of the function are measured vertically. Zeroes are represented as dots positioned on the horizontal axis. The graph either flows from one side of the horizontal axis to the other side as the function changes sign or the graph rebounds from the intercept and stays on the same side of the horizontal axis. The multiplicity of the root tells us which.

Multiplicities

In the graph below, change the sign of exponent for $(x+2)$ from $2$ to $3$.

The sign of $(x+2)^3$ switches across $-2$ and the graph crosses over the axis.

The sign of $(x+2)^2$ stays the same across $-2$ and the graph does not cross over the axis.

If you change the sign of the leading coefficient, then the crossing and bouncing will jump to the other side of the $x$-axis.

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Extrema

Unless we have a quadratic polynomial, the best we can do for the maximums and minimums, at this point, is estimate them.

The graph of $y = f(x) = 0.1(x+2)^3(x-3)$.

$-2$ and $3$ are consecutive zeros or roots of $f$. Since $f$ is continuous, it must remain positive or negative on the interval $(-2,3)$.

This means there must be a local maximum or minimum somewhere inside the interval $(-2,3)$.

This browser does not support embedded elements.

$f(x)$ has an approximate global minimum value of $\answer [tolerance=0.1]{-6.6}$, which occurs approximately at $\answer [tolerance=0.1]{1.75}$.

Calculus will goive us the tools to obtain the derivative for ourselves, and that will help us locate where in the domain maximums and minimums occur.

Rate of Change

Extending our analysis of quadratic functions, the global and local extreme values of a function occur at numbers in the domain where the corresponding points on the graph have horizontal tangent lines. We call these domain numbers critical numbers. Thus, critical numbers play an important role in function analysis.

It will take some further analysis to get a full definition of critical numbers. We’ll start with critical numbers pointing out horizontal tangent lines in the graph and improve from there.

Critical numbers

Polynomials are nice. Their graphs have hills and valleys. Their graphs do not have corners or cusps or vertical tangents.

For polynomials, critical numbers are domain numbers, which correspond to points on the graph where the tangent line has slope $0$ (is horizontal).

Critical numbers are domain numbers marking flat/horizontal places in the graph. These include tops of hills and bottoms of valleys. Critical numbers include places where extreme values occur.

The next example shows that they also include places that are not marking maximums or minimums.

Polynomials are nice. They increase and decrease on intervals defined by critical numbers marking places where the graph is horizontal and possible maximums and minimums.

The graph of $y = f(x) = 0.1(x+2)^3(x-3)$.

This browser does not support embedded elements.

$f$ has two critical numbers: $-2$ and $1.75$. These correspond to the two places where the graph has a horizontal tangent line.

$f$ decreases on $\left (-\infty , \answer {1.75}\right ]$.
$f$ increases on $\left [\answer {1.75}, \infty \right )$.

Note: $-2$ is a critical number, because the graph is flat there. However, the sign of the rate of change of $f$ does not switch there, since $f$ does not have an extreme value there.

Zeros, Factors, and Intercepts

Polynomial functions are nice. They allow us to see many connections that we will keep in mind as we investigate other types of functions.

$\blacktriangleright $ Zeros, Factors, and Intercepts are different things but they all refer to the same information.

If $z_0$ is a zero of the polynomial $p(x)$, means $p(z_0)=0$.
If $(x-z_0)$ is a factor of the polynomial $p(x)$, means $p(x) = q(x) \cdot (x-z_0)$ for some polynomial $q(x)$.
If $(z_0,0)$ is an intercept of the graph of the polynomial $p(x)$, means $(z_0,0)$ is both on the horizontal axis and on the graph of $p$.

However, the existence of any of these implies the other two

If $z_0$ is a zero of the polynomial $p(x)$, then $(x-z_0)$ is a factor and $(z_0,0)$ is on the graph.
If $(x-z_0)$ is a factor of the polynomial $p(x)$, then $p(z_0)=0$ and $(z_0,0)$ is on the graph.
If $(z_0,0)$ is an intercept of the graph of the polynomial $p(x)$, then $p(z_0)=0$ and $(x-z_0)$ is a factor.

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more examples can be found by following this link
More Examples of Poynomial and Rational functions

2026-01-28 00:07:43