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Multiple Roots of Polynomials

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will review the definitions of polynomials and roots of polynomials. We will then look at multiple roots of polynomials and learn some tricks on how to determine a lot about a polynomial by just looking at it in its completely factored form.

Roots of Polynomials

It is said that magicians never reveal their secrets. However, Merle the Math-magician has agreed to let us in on a few of his!

Merle's first trick has to do with polynomials, algebraic expressions which sum up terms that contain different powers of the same variable. He tells us that we will need to know the following facts to understand his trick:

  • A root of a polynomial is a value which, when plugged into the polynomial for the variable, results in 0.
  • A polynomial in completely factored form consists of irreducible factors (meaning they can't be factored any further) which, when multiplied together, give the polynomial.


Polynomials
multroot1


After going over these quick facts, Merle asks a member of the audience to give him a completely factored polynomial off the top of her head. She gives the following polynomial:

  • (x - 1)(x - 1)(x + 3)(x - 5)(x - 5)(x - 5)

Abracadabra! Within seconds, he says the roots of the polynomial are 1, -3, and 5. Some calculations, which are shown on the big screen, verify that Merle is correct!


multroot2


Wow! How did he magically know the roots of the polynomial? He explains that this trick is done using the following theorem:

  • If (x - a) is a factor in the complete factorization of a polynomial, P, then a is a root of P.

Ah-ha! He simply looked at the different factors in the complete factorization and identified a in each one.

  • (x - 1) corresponds to root 1.
  • (x + 3) corresponds to root -3.
  • (x - 5) corresponds to root 5.

What a neat trick! Let's see what Merle has for us next!

Multiple Roots of Polynomials

Merle performs his second trick by predicting the polynomial's graph will cross through the x-axis at x = -3 and x = 5, and will bounce off the x-axis at x = 1. Once again, the big screen displays the graph of the polynomial, and sure enough, he's correct!


multroot3


Impressive! How did he do this one?

Merle tells us that this trick involves being familiar with multiple roots of a polynomial. Multiple roots of a polynomial are roots whose factors show up more than once in the complete factorization of the polynomial. We call the number of times a factor shows up in the complete factorization the multiplicity of the corresponding root.

Consider our polynomial again. The factor (x - 1) shows up in the complete factorization two times, so the multiplicity of the root 1 is 2. Similarly, the multiplicity of -3 is 1, and the multiplicity of 5 is 3.

Notice, we can rewrite the polynomial using powers as follows:

  • (x - 1)2 (x + 3)(x - 5)3

We see that the power of each factor corresponds to the multiplicity of its corresponding root. In general, the root, a, corresponding to the factor (x - a)n, has multiplicity n.

Merle tells us the secret behind his graph trick has to do with the following facts:

  • If a root, a, has an even multiplicity, then the polynomial's graph will bounce off the x-axis at x = a.
  • If a root, a, has an odd multiplicity, then the polynomial's graph will cross through the x-axis at x = a.

Both -3 and 5 have odd multiplicities, so the graph crosses through the x-axis at x = -3 and x = 5, and since 1 has an even multiplicity, the graph bounces off the x-axis at x = 1.

Example

For Merle's final trick, he says he will to turn all of us into math-magicians! To do this, he gives us the following completely factored polynomial:

  • (x + 8)4 (x - 4)5

Now, he asks us: What are the roots of this polynomial, and what will the polynomial's graph do at each of these roots?

We can do this! We'll just use the rules that we've learned!

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