How to Solve Equations that are Not Perfectly Cubed Video

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  • 0:01 Cubic Equations
  • 2:20 The Rational Roots Theorem
  • 2:48 Finding the First Root
  • 5:10 Finding the Other Two Roots
  • 6:34 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.

Watch this video lesson to learn what to do when you have a cubic equation that is not perfectly cubed. Learn an easy method to find all three roots or solutions.

Cubic Equations

Cubic equations, equations with a degree of 3, as you have seen, are found in algebra and other math courses. But, believe it or not, they are not here to make your life harder. They have real-world applications, such as in finding the volume of various objects. Because of this important real-world application and others, mathematicians over the years have developed several methods to solve them. You will learn one of these methods in this video lesson.

To be able to use our method, however, we need to make sure that our cubic equation is written in its standard form of ax^3 + bx^2 + cx + d = 0 where a, b, c and d are numbers and a cannot be 0. Please note that our highest exponent of 3 is written first and then all the other terms are written in descending order based on their exponent. If our cubic equation is not written in standard form, we need to rewrite it so that it is in standard form.

For example, if our cubic equation is given to us as 4x^2 + x^3 + 1 = 0, we need to rewrite it in the proper order with the x^3 term first. So, our cubic equation is rewritten as x^3 + 4x^2 + 1 = 0. We also need to recall our skills in solving quadratic equations, which are equations with a degree of 2. Another skill that we need to recall is that of dividing polynomials. If you feel that you need to refresh your memory on these topics, go ahead and pause this video now while you review these topics.

This method works for any cubic equation that isn't perfectly cubed. What does it mean when a cubic equation isn't perfectly cubed? It means that you don't have a term cubed minus or plus another term cubed. For example, x^3 - 27 = 0 is a perfectly cubed cubic equation. If you had other terms or those terms are not perfect cubes, then you can use this method to solve it. Solving perfectly cubed equations has its own method.

The Rational Roots Theorem

The method that we are going to use is based on the rational roots theorem, which states that possible solutions of a polynomial can be found by dividing a factor of the constant term by a factor of the number associated with the first term. The constant term is the term that is the number by itself with no variable attached to it.

Let's see how we go about using the rational roots theorem to find our first root or solution.

Finding the First Root

We begin with our problem cubic equation. Let's solve the cubic equation, x^3 + 6x^2 + 11x + 6 = 0. We first apply the rational roots theorem to find our first solution. We locate the number associated with our first term, our x^3 term. We see that it is a 1. Our constant term is a 6. Now we need to write out our factors of these numbers. Recall that factors are the numbers that multiply together to get our desired number. The number 1 has only one factor, which is 1. Our 6 has factors of 1, 2, 3 and 6. Now we divide each of our factors of 6 by each of our factors of 1.

Since our number 1 has only one factor, we only need to divide each of our factors of 6 by 1. As you do this, make sure you write both a positive version and a negative version. You can do this by writing the plus or minus symbol in front of each. The plus or minus symbol is the one that has a plus sign on the top of a minus sign. As we divide each of our factors of 6 by our factors of 1, we get this list:

+/- 1, +/- 2, +/- 3, +/- 6

Now that we have our list, we can now begin plugging in various numbers into our cubic equation. Our goal is to find a number that makes our equation equal to 0. How do we pick a number to try? Well, we can look at our equation to see if it will give us a clue. We see that all of the signs of our equation are pluses, so that tells me that I need to plug in a negative number so that we end up with both positive and negative numbers that will add up to 0.

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