Polynomials and their roots

Group activity: Give a definition of polynomial.

The idea of polynomials

Polynomials are the (smallest) collection of functions satisfying:

  • The constant functions and the identity function f(x)=x are polynomials.
  • If you combine two polynomials using the operations addition, subtraction, or multiplication, then the result is a polynomial.

The definition of polynomials

Definition: a polynomial function is any function of the form f(x)=anxn+an−1xn−1+⋯+a1x+a0 for fixed numbers an,…,a0 (an≠0) and n is called its degree.

Group activity:

  • What do the following terms mean with regards to polynomials?
    • degree
    • root
    • factor
  • Describe any connections between these three terms.

Pre-discussion poll of the class

  • Can all polynomials be factored?
  • Does every polynomial have a root?
  • If a polynomial has a root at x=2, does the graph of the polynomial cross the x-axis at x=2?
  • If a polynomial has a factor, does it have a root?
  • If a polynomial has a root, does it have a factor?
  • Is it possible to factor a polynomial that has no root?
  • A polynomial of degree n has _______ roots.

Examples. Look at the graph of each polynomial. How many roots does it have? Does it factor?

  • f(x)=x2−5x+6 (2 factors, 2 roots)
  • g(x)=x2−3x+4 (no factors, no roots)
  • h(x)=x3−9x2+27x−27 (3 factors, 1 root)
  • j(x)=x3−x (3 factors, 3 roots)
  • k(x)=x4+3x2+2 (2 quadratic factors, no roots)

Theorem. If a polynomial f(x) has a root r, then f(x) can be written in the form f(x)=(x−r)⋅q(x), where q(x) has degree one less than the degree of f(x).

What does this theorem allow us to say about the roots of a polynomial f(x) of degree 4? Discuss

Theorem. Every polynomial can be factored into a product of linear and irreducible quadratic factors.

Multiplicity and the shape of the graph near a root

Definition. The multiplicity of a root x=r in a polynomial is the number of times that the linear factor (x−r) appears in the complete factorization of that polynomial.

Shape of the graph near a root. If a polynomial has a root at x=r, the behavior of the graph near r always fits one of these descriptions:

  1. The graph passes through the x-axis at x=r without leveling off.
  2. The graph levels off until it is tangent to the x-axis at x=r. It touches but does not cross the x-axis there.
  3. The graph levels off until it is tangent to the x-axis at x=r, and crosses the x=axis there.

Exercise: For each polynomial, what are the roots? What is the multiplicity of each root? What is the shape of the graph near the root?

  • f(x)=x
  • g(x)=x2
  • h(x)=x3
  • j(x)=x−1
  • k(x)=(x−2)4
  • m(x)=(x−3)7

Exercise. For each polynomial, what are the roots? What is the multiplicity of each root? What is the shape of the graph near the root?

  • h(x)=x3−1
  • f(x)=x5(x2+2x+1)(x2+2x+6)
  • g(x)=(x−5)(x+2)(x−6)(x+2)(x−6)(x−5)(x−6)(x+2)

Question: What is the connection between the multiplicity of a root and the behavior the graph near that root?

Exercise. Guess the formula for each polynomial. You may use Desmos or another graphing utility.

a.

Polynomial with roots at -3, 1, and 5

b.

Polynomial with roots at 2, -2

c.

Polynomial with roots at -4, ..., 4

d.

Polynomial approximating cosine x

Theorem. If a polynomial f(x) has odd degree, then it has at least one root.

Theorem. The end behavior (the limit as approaches infinity or -infinity) of a polynomial is the same as that of the leading term anxn.

Resource: For a more detailed exposition of this material, here is a page about Polynomials and their graphs on the MAT 1375 Course Hub (it includes a great video with examples of finding polynomials to match polynomial descriptions, created by past City Tech MEDU student Irania Vasquez)