Solving Higher Degree Polynomials

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  • 0:02 Higher Degree Polynomials
  • 0:47 Rational Roots Theorem
  • 1:57 Possible Solutions
  • 3:42 Finding the Other Solutions
  • 5:27 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will be able to solve polynomials where the degree is three or higher. Learn the technique that you can use to solve any one of these polynomials.

Higher Degree Polynomials

So, right now you are familiar with linear equations, where we have variables with no exponents, and you are familiar with quadratic equations, where the highest exponent is a 2. Now we are moving on to the next one up. This group actually covers all the higher-up polynomials, so it covers the polynomials of degree 3 and higher. We call these higher degree polynomials.

Recall that the degree of any polynomial is the highest exponent in that polynomial. Because we are dealing with higher degree polynomials, we will use a different technique to solve them. This technique allows us to find our solutions one by one.

Rational Roots Theorem

This technique that we will use is called the rational roots theorem. It tells us that the possible solutions of the polynomial can be found in the list of numbers generated by putting the factors of the last constant term over the factors of the leading coefficient.

We end up with a list of fractions. These fractions can sometimes be simplified into whole numbers. We have to remember that this is only a list of possible answers. It does not give us a list of solutions. In fact, it is possible that this theorem doesn't provide any solutions at all. But, since math problems in textbooks and on tests are created to help you learn, you most likely will not come across such a scenario. You will most likely find your solutions from the list. This theorem gives you rational solutions; it won't give you imaginary solutions or complex solutions. You will thus see that the problems will ask you to find the rational solutions to a polynomial.

Let's take a look at how we use this theorem.

Possible Solutions

Let's try to find the rational solutions to this polynomial:

higher degree polynomial

First, we will create our list of possible solutions. We list our possible factors of our last constant term, -24, over the factors of our leading coefficient, 1. Our factors of -24 are +/- 1, 2, 3, 4, 6, 8, 12, 24. Yes, we have a positive and negative version for each factor. This is because we can multiply with either the positive or negative version to get 24. For example, we can do 2 * -12 or -2 * 12 to get to -24.

See how we have both the positive and negative versions covered? Our leading coefficient is 1, so our factors here are +/- 1. Again, we have both positive and negative since we can do either 1 * 1 or -1 * -1 to get to 1.

Now that we have our factors, let's find our fractions. We have +/- 1/1, 2/1, 3/1, 4/1, 6/1, 8/1, 12/1, 24/1. Because our denominator is 1, we can simplify our list to +/- 1, 2, 3, 4, 6, 8, 12, 24. Now we have our complete list of possible rational solutions.

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