2.12: Real Zeros of Polynomials
In the real world, problems do not always easily fit into quadratic or even cubic equations. Financial models, population models, fluid activity, etc., all often require many degrees of the input variable in order to approximate the overall behavior. While it can be challenging to model some of these more complex interactions, the effort can be well worth it. Mathematical models of stocks are used constantly as a way to "look into the future" of finance and make the kinds of educated guesses that are behind some of the largest fortunes in the world.
What benefits can you think of to modeling the behavior of large populations? Can you think of other useful applications not mentioned here?
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 PatrickJMT: Finding all Zeroes of a Polynomial Function
Guidance
There are three theorems and a rule that we will be referring to during this lesson in order to help make the discovery of the roots of polynomial functions easier. You should review them and be prepared to refer to them often during the practice problems.
The Remainder Theorem
If a polynomial of degree is divided by , then the remainder is a constant and it is equal to the value of the polynomial when is substituted for . That is
The Factor Theorem
If is a polynomial of degree and , then is a factor of the polynomial . Further, if is a factor, then is a zero of .
The Rational Zero Theorem
Given the polynomial
and is a positive integer. If the coefficients are integers and is a rational zero in lowest terms, then is a divisor of and is a divisor of .
Descartes Rule of Signs
Given any polynomial, ,
 Write it with the terms in descending order, i.e. from the highest degree term to the lowest degree term.
 Count the number of sign changes of the terms in . Call the number of sign changes .
 Then the number of positive roots of is less than or equal to .
 Further, the possible number of positive roots is
 To find the number of negative roots of , write in descending order as above (i.e. change the sign of all terms in with odd powers), and repeat the process above. Then the maximum number of negative roots is .
Example A
Use synthetic division and the remainder and factor theorems to find the quotient and the remainder if is divided by .
Solution
Hence
Notice that the remainder is 181 and it can also be obtained if we simply substituted into ,
Example B
Use the rational zero theorem and synthetic division to find all the possible rational zeros of the polynomial
Solution
From the rational zero theorem, is a rational zero of the polynomial . So is a divisor of 2 and is a divisor of 1. Hence, can take the following values: 1, 1, 2, 2 and can be either 1 or 1. Therefore, the possible values of are
So there are four possible zeros. Of these four, not more than three can be zeros of because is a polynomial with degree 3. To test which of the four possible candidates are zeros of , we use the synthetic division. Recall from the remainder theorem if , then is a zero of . We have
Hence, 2 is a zero of . Further, by the division algorithm,
The remaining zeros of are simply the zeros of which is easier to manipulate,
and thus the remaining zeros are 1 and 1. Thus the rational zeros of are 1, 1, and 2.
Example C
Graph the polynomial function .
Solution
Notice that the leading term is , where odd and . This tells us that the end behavior will take the shape of a power function with an odd exponent.
Here, as you can see, there is no straightforward way to find the zeros of . However, with the use of the factor theorem and the synthetic division, we can find the rational roots of .
First, we use the rational zero theorem and find that the possible rational zeros are
testing all these numbers by the synthetic division,
1 is not a root. Now let's test .
we find that 1 is a zero of and so we can rewrite ,
Looking at quadratic part,
and so
Thus 1 and are the intercepts of . The intercept is
Further, the synthetic division can be also used to form a table of values for the graph of :
We choose test points from each interval and find .
Interval  Test Value  Sign of  Location of points on the graph  

1  28    below the axis  
  below the axis  
3  4  +  above the axis 
From this information, the graph of is shown in the two graphs below. Notice that the second graph is a magnification of in the vicinity of the axis.
Were you able to identify some valuable realworld uses for modeling higherdegree polynomials? Here are a few possibilities:
There are many, many more. 

Guided Practice
Questions:
1) Show that is a factor of .
 Find the quotient and express in factored form.
2) Use the 'rational zero' theorem and synthetic division to find all the possible rational zeros of the polynomial
3) Use Descartes Rule of Signs to identify the possible number of positive and negative roots of
 .
4) Find the root(s) of
5) Find the root(s) of
Solutions
1) By the factor theorem, if , then is a factor of the polynomial. In other words, if the synthetic division produces a remainder equal to zero, then is a factor of the polynomial. :Using the synthetic division with ,
 Hence, , and the quotient is
 so that can be written as
2) Assume is a rational zero of . By the rational zero theorem, is a divisor of 6 and is a divisor of 1. Thus and can assume the following respective values

 and
 Therefore, the possible rational zeros will be
 Notice that with these choices for and there could be rational zeros. But, eight of them are duplicates. For example . The next step is to test all these values by the synthetic division (we'll let you do this on your own for practice) and we finally find that
 are zeros of . That is
3) First, rewrite in descending order
 The number of sign changes of is 2, so the number of positive roots is either 2 or 0.
 For the negative roots, write
 The number of sign changes of is 2, so the maximum number of negative roots is 2.
 The graph of below shows that there is one negative root and two positive roots.
4) Solve by factoring and applying the zero product rule:
 or
 or
 (each zero has a multiplicity of one)
5) This one is easy:
 Since there are no real roots of even powers, this function has zero real solutions.
Practice
Problems 1  3: Use a) long division and b) synthetic division to perform the divisions.
 Express each result in the form:
 by
 by
 by
 Use synthetic division to find and so that if and
 If , use synthetic division to determine the following: a) b) c) d) e) What are the factors of ?
 Find so that is a factor of
 Use synthetic division to determine all the zeros of the polynomials: a) b)
 Graph the polynomial function by using synthetic division to find the intercepts and locate the intercepts.
 Graph the polynomial function by using synthetic division to find the intercepts and locate the intercepts.
 Write a 3rd degree equation of a polynomial function with the zeroes: 0, 2, and 5.
 Write a 7th degree equation of a polynomial function with the zeroes: 0 (multiplicity 2), 2 (multiplicity 3), and 5 (multiplicity 2)
 Write a quadratic equation which has 4 (multiplicity 2) as the zero and opens downward.
 Write a 3rd degree polynomial function with the zeroes: 2, 2, and 6, passing through the point (3, 4)
 Let and find the solutions: a) b)
 Graph and find the solution set of the inequality .
 Use the graph of to find the solution set of the inequality .
Descartes' Rule of Signs
Descartes' rule of signs is a technique for determining the number of positive and negative real roots of a polynomial.factor theorem
The factor theorem states that if is a polynomial of degree and , then is a factor of the polynomial .factorization theorem
The factorization theorem states that If , where , and is a positive integer, then where the numbers are complex numbers.Multiplicity
The multiplicity of a term describes the number of times the given term acts as a zero of the given function.Polynomial
A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.Rational Zero Theorem
The rational zero theorem states that for a polynomial, , where are integers, the rational roots can be determined from the factors of and . More specifically, if is a factor of and is a factor of , then all the rational factors will have the form .Remainder Theorem
The remainder theorem states that if , then is the remainder when dividing by .Roots
The roots of a function are the values of x that make y equal to zero.Synthetic Division
Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.Zeroes
The zeroes of a function are the values of that cause to be equal to zero.Zeros
The zeros of a function are the values of that cause to be equal to zero.Image Attributions
Description
Learning Objectives
Here you will learn about using polynomial long division and synthetic division to find roots, and how to apply synthetic division to find the rational zeros of an unfactored polynomial (of degree >2).
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Date Created:
Nov 01, 2012Last Modified:
Jun 08, 2015Vocabulary
Descartes' Rule of Signs
Descartes' rule of signs is a technique for determining the number of positive and negative real roots of a polynomial.factor theorem
The factor theorem states that if is a polynomial of degree and , then is a factor of the polynomial .factorization theorem
The factorization theorem states that If , where , and is a positive integer, then where the numbers are complex numbers.Multiplicity
The multiplicity of a term describes the number of times the given term acts as a zero of the given function.Polynomial
A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.Rational Zero Theorem
The rational zero theorem states that for a polynomial, , where are integers, the rational roots can be determined from the factors of and . More specifically, if is a factor of and is a factor of , then all the rational factors will have the form .Remainder Theorem
The remainder theorem states that if , then is the remainder when dividing by .Roots
The roots of a function are the values of x that make y equal to zero.Synthetic Division
Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.Zeroes
The zeroes of a function are the values of that cause to be equal to zero.Zeros
The zeros of a function are the values of that cause to be equal to zero.