Polynomials, Factors, and Zeros
 Home > Lessons > Polynomials, Factors, and Zeros Search | Updated September 27th, 2020
Introduction
In this section, you will learn how a polynomial's zeros, factors, and graph are all related. Here are the sections within this lesson:

So that you will be able to understand the lessons that follow, we need to review a few definitions.

 polynomial A polynomial is a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables each raised to nonnegative integral powers, such as…3x2 + 2x - 5. zero A zero is the location where a polynomial intersects the x-axis. These locations are called zeros because the y-values of these locations are always equal to zero. factor A factor is one of the linear expressions of a single-variable polynomial. A polynomial can have several factors, such as the factors...(x - 1) and (x + 3).

When polynomials are graphed, many of them intersect the x-axis. The locations where a polynomial crosses the x-axis are called ‘zeros.’ For instance, we can graph this polynomial.

This is the graph of the polynomial.

Notice how the polynomial intersects the x-axis at two locations. It intersects at (-4, 0) and (2, 0). So, we say the polynomial has zeros at -4 and 2.

Likewise, we can graph this polynomial.

This is the graph of the polynomial.

It can easily be seen that this polynomial has three zeros, namely -1, 0, and 1.

The factors of a polynomial are important to find because they can be multiplied together to gain a polynomial.     In our lesson on zeros, we saw this graph.

We saw that the zeros were -4 and 2. To get the factors, we simply take the opposite of the zeros. These are the factors.

We also saw this graph in the last section.

It’s zeros are -1, 0, and 1. Here are it’s factors.

The (x – 0) factor is the same thing as writing x. So, here are the factors written more properly.

Or, we can write them more compactly as…

We have an entire section devoted to operations on polynomials, which will come in handy for our next section. You can find it here: Operations on Polynomials. Please review the section before proceeding to the next section.

Besides learning how to add, subtract, multiply and divide polynomials, it is extremely convenient to have a graphing calculator. Modern graphing calculators, like the TI-Nspire CX CAS model, are excellent tools for quickly displaying graphs and consequently speed up the learning process.

We will examine the connection between a polynomial's zeros, factors, and expanded forms.

Let's again revisit this polynomial as our first example.

If we look at the graph of this polynomial, we get this picture.

It has the zeros at -4 and 2, which means it has the factors (x + 4) and (x - 2).

If we were to multiply the factors (x + 4) and (x - 2), this is what we would get.

This simplifies as follows.

The above polynomial is the original polynomial. This demonstrates how the graph of a polynomial is related to its zeros and its factors.

For our second example, we should look at the cubic polynomial we saw in a previous section.

Viewing this polynomial on a coordinate plane yields this picture below.

It has zeros at -1, 0, and 1, which means its factors are...

Or, we can write them as...

If we multiply the (x + 1) and the (x - 1), we get this.

This simplifies as follows.

Now, we have to multiply this result by the last factor, x.

This yields our final result and our original polynomial.

This indicates how a polynomial's expanded form is related to its zeros and its factors.

This video explains how to use complex zeros to gain factors, and a polynomial expression. It also will inform you of complex conjugate pairs.

When graphing polynomials, there are patterns to follow. These activities and quizzes will assist the learning process to graph polynomial equations.

ctivity: Translation: Discovery
ctivity: Translating Polynomials

Like there are logical rules that tell us the connections between zeros and factors, there are also rules that govern the end behavior (branches) of a polynomial. Knowing what the y-values are when the x-values are very large and very small will help us sketch the graphs of polynomials.

[Let it be known there is a big difference between even and odd degree and even and odd functions.]

This video will instruct you on the connections between a polynomial's degree, leading coefficient, and its end behavior.

ideo: Even and Odd Degree

When we deal with polynomials, two terms emerge: root and zero. There is a distinction between the two even though some people use them interchangeably. Here are their differences:

 root It is the solution of an equation. 2 and -2 are roots ofx2 - 4 = 0 because 2 and -2 are solutions to the equation. zero It is a value for which a function is equal to zero. 5 is a zero forf(x) = x - 5 becausef(5) = 0.

The Rational Root Theorem is an involved concept. Understanding it and using requires several layers of explanation, which is done in this video.

Try our instructional videos that detail the lessons above.

Try our interactive quizzes to determine if you understand the lessons above.

Use these activities to learn about polynomials, factors, and zeros.

ctivity: Factors and Zeros
ctivity: Translation: Discovery
ctivity: Translation: Discovery
ctivity: Translating Polynomials

Try these lessons, which are closely related to the lessons above.

esson: End Behavior
esson: Operations on Polynomials
esson: Factoring Trinomials
esson: Polynomial Models
esson: PARCC Problems and Solutions
esson: The Rational Root Theorem