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

Learning Outcomes.

  • 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:

  1. Polynomials – Theory
  2. 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 x-c of that polynomial. However, to understand it, we need to work with complex numbers. Here’s a brief reminder:

Definition. The imaginary unit i satisfies i^2=-1, and so i=\sqrt{-1}.

Definition. A complex number has the form a+bi, where a and b are real numbers and i 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 f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} be a non-constant
polynomial. Then there exists a complex number c which is a root of f.

Notes:

  • The theorem doesn’t say anything about what kind of coefficients f has — in particular, they can be complex (or real, or rational, or integer) numbers.
  • The function may not have any real roots, for example f(x)=x^2+1. However, it will have complex roots (in this case c=i and c=-i).
  • Once we find a root c, we can use polynomial division to factor f(x)=q(x)(x-c).

Example. Find roots of the given polynomial and use this information to factor the polynomial completely.
a) f(x)=2 x^{3}-8 x^{2}-6 x+36
b) f(x)=x^{4}-3 x^{3}-36 x^{2}+68 x+240
c) f(x)=x^{3}+1
d) f(x)=x^{4}-16

VIDEO: Fundamental Theorem of Algebra – Examples 1

Observations

  • Every polynomial f(x) of degree n can be factored in this form:

        \[f(x)=m \cdot\left(x-c_{1}\right) \cdot\left(x-c_{2}\right) \cdots \cdots\left(x-c_{n}\right)\]

  • Thus every polynomial of degree n has at most n roots.
  • A factor x-c may appear more than once. The number of times that x-c appears as a factor is called the multiplicity of the corresponding root c
  • A nice fact: If a polynomial f(x) has real coefficients, and has a complex number a+bi as a root, then f(x) also has the complex conjugate a-bi as a root.

Example. Find a polynomial f with the following properties.
a) f has degree 3, the roots of f are precisely 4,5,6, and the leading coefficient of f is 7
b) f has degree 3 with real coefficients, f has roots 3 i,-5 (and possibly other roots as well), and f(0)=90
c) f has degree 4 with complex coefficients, f has roots i+1,2 i, 3

VIDEO: Fundamental Theorem of Algebra – Example 2

The rational root theorem

Observation (Rational root theorem). Consider the equation

    \[a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}=0\]


where every coefficient a_{n}, a_{n-1}, \ldots, a_{0} is an integer and a_{0} \neq 0, a_{n} \neq 0. Assume that x=\frac{p}{q} is a solution and the fraction x=\frac{p}{q} is completely reduced. Then a_{0} is an integer multiple of p, and a_{n} is an integer multiple of q.

Therefore, all possible rational solutions are fractions x=\frac{p}{q} where p is a factor of a_{0} and q is a factor of a_{n}. 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 \frac{1}{2} or -7. A polynomial may have irrational roots, like \sqrt{2}, 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 x=\frac{p}{q} where p is a factor of a_{0} (that is, p goes into a_0 evenly) and q is a factor of a_{n}.

Example.
a) Find all rational roots of f(x)=7 x^{3}+x^{2}+7 x+1
b) Find all real roots of f(x)=2 x^{3}+11 x^{2}-2 x-2
c) Find all real roots of f(x)=4 x^{4}-23 x^{3}-2 x^{2}-23 x-6

VIDEO: Rational Root Theorem – Examples

Exit Question

Find a polynomial f of degree 5 with real coefficients, the leading coefficient is 1, and the roots are determined by its graph:

Graph of polynomial function f(x)

Answer

f(x)=(x-1)(x-2)(x-3)(x-4)^{2}