﻿ Introduction to Polynomials

Introduction to Polynomials

First, review the Introduction to Polynomials lesson in the Algebra II curriculum to make sure you've mastered basic polynomial concepts and terminology.

For example, you should know:

The Definition of a Polynomial
A polynomial is a finite sum of terms, each of the form $\,ax^k\,,$ where $\,a\,$ is a real number, and $\,k\,$ is a nonnegative integer. That is: $\,k\in \{0,1,2,3,\ldots\}$
How to Put a Polynomial in Standard Form
The standard form of a polynomial is: $$\cssId{s11}{a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1 + a_0}$$ Here, $\,n\,$ denotes the highest power to which $\,x\,$ is raised.

Thus, in standard form, the highest power term is listed first, and subsequent powers are listed in decreasing order.

How to Determine a Polynomial's Degree
The degree of the polynomial is the highest power to which $\,x\,$ is raised.
How to Determine a Polynomial's Leading Coefficient
The coefficient of the highest power term is called the leading coefficient of the polynomial.

When a polynomial is written in standard form, the leading coefficient actually leads the polynomial.

For example, $$P(x) = -3x + 11 - 5x^7 + 9x^2$$ is a polynomial.
Its standard form is: $$P(x) = -5x^7 + 9x^2 - 3x + 11$$

Its degree is $\,7\,.$ Its leading coefficient is $\,-5\,.$

For a more thorough review and practice exercises, study Introduction to Polynomials in the Algebra II curriculum.

Relationship between the Zeros and Factors of a Polynomial

There is a beautiful relationship between the zeros and factors of a polynomial, which is explored in the next section. In preparation, the concepts of zero and factor are reviewed below.

Zeros (Roots) of Functions

It would be reasonable to guess that a zero of a function has something to do with the number $\,0\,.$ It does!

What input (or inputs) must be dropped in the top of a function box, to get zero out the bottom?

A zero of a function is an input whose corresponding output is zero.

Example (Finding the Zeros of a Function)

Consider the function $\,f(x) = x^2 - 2x + 1\,.$ Recall that $\,f(x)\,$ represents the output from the function $\,f\,$ when the input is $\,x\,.$

 What value(s) of $\,x\,$ make $\,f(x)\,$ equal to $\,0\,$? $f(x) = 0$ Set the output, $\,f(x)\,,$ equal to zero. $x^2 - 2x + 1 = 0$ Substitution: $\,f(x)\,$ is $\,x^2 - 2x + 1\,$ $(x-1)^2 = 0$ Factor: $\,x^2 - 2x + 1 = (x-1)^2$ $x-1 = 0$ $z^2 = 0\,$ if and only if $\,z = 0$ $x = 1$ Add $\,1\,$ to both sides.

In this example, there is only one input, $\,x = 1\,,$ for which the output is zero.

Check:

$$\cssId{s54}{f(1) = 1^2 - 2(1) + 1 = 0}$$

Here, $\,1\,$ is the only zero of the function $\,f\,.$

A zero of a function is also called a root of the function.

Initially, you might find yourself a bit uncomfortable saying ‘$\,1\,$ is a zero’. It almost sounds like you're saying ‘$\,1 = 0\,$’, which is of course false. The word ‘a’ is critically important in the sentence!

 When you say: $\,1\,$ is a zero You're saying: $\,1\,$ is an input whose corresponding output is zero

Example (Zeros of Non-Polynomials)

You can talk about zeros of any function—not just polynomials.

For example, the number $\,2\,$ is a zero of $\,f(x) = \sqrt{x - 2}\,$ since $\,f(2) = \sqrt{2-2} = 0\,.$

The function $\,f(x) = {\text{e}}^x\,$ doesn't have any zeros, since $\,{\text{e}}^x\,$ is always strictly greater than zero.

Equivalent Characterizations of a Zero of a Function

There are several equivalent ways to think about zeros of functions:

ZEROS (ROOTS) OF FUNCTIONS

A zero (or root) of a function is an input, whose corresponding output is zero.

Let $\,f\,$ be a function, and let $\,c\,$ be an input to $\,f\,.$

Then, the following are equivalent:

• $c\,$ is a zero of $\,f$
• $c\,$ is a root of $\,f$
• $c\,$ is an input, with corresponding output $\,0$
• $f(c) = 0$
• the point $\,(c,0)\,$ lies on the graph of $\,f\,$
• $f\,$ has an $x$-intercept equal to $\,c$
• the graph of $\,f\,$ crosses the $x$-axis at $\,c\,$
• $x = c\,$ is a solution of the equation $\,f(x) = 0$

In particular, observe that zeros are easy to spot if you have the graph of a function: they are the $x$-intercepts.

Review: Factors of a (Whole) Number

Early on, you learned that the factors of (say) $\,27\,$ are numbers that go into $\,27\,$ evenly (with no remainder). ‘No remainder’ means the same thing as a remainder of zero.

For example, $\,3\,$ is a factor of $\,27\,.$ Why? Because it goes into $\,27\,$ nine times:  $27 = 3\cdot 9$

The number (say) $\,11\,$ is not a factor of $\,27\,.$ Why not? Because there is a nonzero remainderit goes in twice, with a remainder of $\,5\,.$ That is:  $\,27 = 2\cdot 11 + 5$

Note that when you write $\,27 = 3\cdot 9\,,$ the number $\,27\,$ has been expressed as a product (the last operation is multiplication). However, when you write $\,27 = 2\cdot 11 + 5\,,$ the number $\,27\,$ has been expressed as a sum (the last operation is addition).

Key Idea:  The factors of a number can be used to express the number as a product. Indeed, factoring means to take something and express it as a product.

Factors of a Polynomial

In a similar way, we can talk about factors of a polynomial. These are expressions that go into the polynomial evenly (with no remainder). (Note: long division of polynomials is covered in a future section.)

For example, let: $$P(x) = x^2 + x - 6$$

The expression $\,x-2\,$ is a factor of $\,P(x)\,.$ Why? Because it goes into $\,P(x)\,$ evenly:  it goes in $\,x + 3\,$ times.

That is: $$x^2 + x - 6 = (x-2)(x+3)$$

Check by multiplying out:

\begin{align} &(x-2)(x+3)\cr &\qquad = x^2 + 3x - 2x - 6\cr &\qquad = x^2 + x - 6 \end{align}

The expression (say) $\,x+5\,$ is not a factor of $\,P(x)\,.$ Why not? Because when $\,x+5\,$ is divided into $\,P(x)\,,$ there is a nonzero remainder.

Indeed, long division of polynomials shows that $\,x+5\,$ goes into $\,P(x)\,$ $\,x-4\,$ times, with a remainder of $\,14\,.$ That is:

$$\cssId{s124}{x^2 + x - 6 = (x+5)(x-4) + 14}$$

Check by multiplying out and adding:

\begin{align} &(x+5)(x-4) + 14\cr &\qquad = x^2 - 4x + 5x - 20 + 14\cr &\qquad = x^2 + x - 6 \end{align}

Note that when you write $\,P(x) = (x-2)(x+3)\,,$ the polynomial $\,P(x)\,$ has been expressed as a product (the last operation is multiplication).

However, when you write $\,P(x) = (x+5)(x-4) + 14\,,$ the polynomial $\,P(x)\,$ has been expressed as a sum (the last operation is addition).

Key Idea:  The factors of a polynomial can be used to express the polynomial as a product.

Looking Ahead to the Next Section...

Expressions have lots of different names, and different names are good for different things.

When you write $\,P(x) = x^2 + x - 6\,,$ it's not the least bit clear (at least to this author) what value(s) of $\,x\,$ make the output zero. What number, when squared, then added to itself, then with $\,6\,$ subtracted, gives zero? Not a clue!

However, when you write $\,P(x) = (x-2)(x+3)\,,$ the zeros of $\,P\,$ jump out at you!

The number $\,2\,$ is a zero:

\begin{align} P(2) &= (2-2)(2+3)\cr &= 0\cdot 5\cr &= 0 \end{align}

The number $\,-3\,$ is a zero:

\begin{align} P(-3) &= (-3-2)(-3+3)\cr &= -5\cdot 0\cr &= 0 \end{align}

Returning to the other name (just for fun):

\cssId{s146}{ \begin{align} P(2) &= 2^2 + 2 - 6\cr &= 4 + 2 - 6\cr &= 0 \end{align}} \cssId{s147}{ \begin{align} P(-3) &= (-3)^2 + (-3) - 6\cr &= 9 - 3 - 6\cr &= 0 \end{align}}

You may have noticed the beautiful relationship between zeros and factors:

$x-\color{blue}{2}$ is a factor;
$\color{blue}{2}\,$ is a zero

$x + 3 = x-(\color{blue}{-3})$ is a factor;
$\color{blue}{-3}\,$ is a zero

This observation is formalized in the next section!