Polynomials, Factors, and Zeros
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    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.

polynomialA 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.
zeroA 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.
factorA 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…

    uiz: Finding Zeros and Factors, Given a Polynomial

    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.

    Use this video and quiz to learn more and test your understanding.

    ideo: Real Zeros to Factors to Polynomials
    uiz: Finding a Polynomial, Given its Real Zeros

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

    ideo: Complex Zeros to Factors to Polynomials
    uiz: Finding a Polynomial, Given its Imaginary Zeros

    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

    uiz: Translations: From Equations to Statements
    uiz: Translations: From Statements to Equations

    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:

rootIt is the solution of an equation.2 and -2 are roots of
x2 - 4 = 0 because 2 and -2 are solutions to the equation.
zeroIt is a value for which a function is equal to zero.5 is a zero for
f(x) = x - 5 because
f(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.

    ideo: The Rational Root Theorem

    Try our instructional videos that detail the lessons above.

    ideo: Polynomials: Factors & Zeros
    ideo: Real Zeros to Factors to Polynomials
    ideo: Complex Zeros to Factors to Polynomials
    ideo: Multiplying Polynomials
    ideo: Dividing Polynomials: Synthetic Division
    ideo: Even and Odd Degree
    ideo: The Rational Root Theorem

    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