Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.
Lesson 10: Roots of polynomials
Topic. This lesson covers Session 10: Roots of polynomials
- Understand the relationship between factors and roots of polynomials as described in the Fundamental Theorem of Algebra, and apply it to solve problems.
- Understand the hypotheses and conclusions of the Rational Roots Theorem, and apply it to solve problems.
WeBWorK. There are two WeBWorK assignments on today’s material:
- Polynomials – Theory
- Polynomials – Rational Roots
Additional Video Resources.
Question of the Day: What is the connection between the zeros of a polynomial and the factors of the polynomial?
Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra describes exactly the connection between the roots of a polynomial and the linear factors of that polynomial. However, to understand it, we need to work with complex numbers. Here’s a brief reminder:
Definition. The imaginary unit satisfies , and so .
Definition. A complex number has the form , where and are real numbers and is the imaginary unit.
For a review of complex numbers, take a look at this online lesson on Complex Numbers from the MAT 1275 Course Hub.
Fundamental Theorem of Algebra
Let be a non-constant
polynomial. Then there exists a complex number which is a root of .
- The theorem doesn’t say anything about what kind of coefficients has — in particular, they can be complex (or real, or rational, or integer) numbers.
- The function may not have any real roots, for example . However, it will have complex roots (in this case and ).
- Once we find a root , we can use polynomial division to factor .
Example. Find roots of the given polynomial and use this information to factor the polynomial completely.
VIDEO: Fundamental Theorem of Algebra – Examples 1
- Every polynomial of degree can be factored in this form:
- Thus every polynomial of degree has at most roots.
- A factor may appear more than once. The number of times that appears as a factor is called the multiplicity of the corresponding root
- A nice fact: If a polynomial has real coefficients, and has a complex number as a root, then also has the complex conjugate as a root.
Example. Find a polynomial with the following properties.
a) has degree the roots of are precisely and the leading coefficient of is 7
b) has degree 3 with real coefficients, has roots (and possibly other roots as well), and
c) has degree 4 with complex coefficients, has roots
VIDEO: Fundamental Theorem of Algebra – Example 2
The rational root theorem
Observation (Rational root theorem). Consider the equation
where every coefficient is an integer and . Assume that is a solution and the fraction is completely reduced. Then is an integer multiple of , and is an integer multiple of .
Therefore, all possible rational solutions are fractions where is a factor of and is a factor of . We can use this observation to find good candidates for the roots of a given polynomial.
What’s the big deal?
Why do we care?
- The rational root theorem does something extremely nice – if we are searching the number line for roots of a polynomial, it narrows down the search from the entire number line to just a few points. We can’t test every number in the number line, but we can test just a few points.
What are the restrictions?
- We have to make sure our polynomial only has integer coefficients (just look at the formula and make sure all the numbers you see are whole numbers, positive or negative).
What kind of roots will we find?
- The theorem only finds roots that are rational numbers, like or . A polynomial may have irrational roots, like , which we will have to find using other methods.
How do we use the Rational Root Theorem?
- We can find all rational roots by considering fractions where is a factor of (that is, goes into evenly) and is a factor of .
a) Find all rational roots of
b) Find all real roots of
c) Find all real roots of
VIDEO: Rational Root Theorem – Examples