Sum and Difference of Cubes

Jun 04, · We’ll know when we have a sum of cubes because we’ll have two perfect cubes separated by addition. When that’s the case, we can take the cube (third) root of each term and use a formula to factor. The formula for the sum of two cubes is???a^3+b^3=(a+b)(a^2-ab+b^2)??? The key is to “memorize” or remember the patterns involved in the formulas. {a^3} + {b^3} a3 + b3 is called the sum of two cubes because two cubic terms are being added together. {a^3} - {b^3} a3 ? b3 is called the difference of two cubes because two cubic terms are being subtracted.

I create online courses to help you rock your math class. Read more. Both terms are perfect cubes, so we can use the cube plus a cube formula to factor. The formula is. In this case??? We can check our work by distributing each term in the binomial factor over each term in the trinomial factor. Both terms are perfect cubes, so we can use the formula for factoring the sum of perfect cubes.

How to factor the sum of two how to setup proxy server in internet explorer. I'm krista. The formula for the sum of two cubes is??? Take the course Want to learn more about Algebra 2? I have a step-by-step course for that. Learn More. Factoring the sum of two perfect squares, step-by-step Example Factor the polynomial.

First check to see if each term is a perfect cube. The formula is??? Example Factor the expression. In this case,??? We use the sum of cubes formula to get??? Get access to the complete Algebra 2 course. Get started. Learn math Krista King June 4, mathlearn onlineonline courseonline mathalgebraalgebra 2algebra iifactoringsum of cubesfactoring the sum of cubesperfect cubestwo cubestwo perfect cubes.

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The formula we use to factor a binomial which is the sum of two perfect cubes

Feb 06, · ?? Learn how to factor polynomials using the sum or difference of two cubes. A polynomial is an expression of the form ax^n + bx^(n-1) + + k, where a. The sum of two cubes equals the sum of its roots times the squares of its roots minus the product of the roots, which looks like Like the results of factoring the difference of two cubes, the results of factoring the sum of two cubes is also made up of: A binomial factor (a + b), and. Sum and Difference of Cubes. The sum or difference of two cubes can be factored into a product of a binomial times a trinomial. That is, x 3 + y 3 = (x + y) (x 2 ? x y + y 2) and x 3 ? y 3 = (x ? y) (x 2 + x y + y 2). A mnemonic for the signs of the factorization is the word "SOAP", the letters stand for "Same sign" as in the middle of the original expression, "Opposite sign", and "Always Positive".

The other two special factoring formulas you'll need to memorize are very similar to one another; they're the formulas for factoring the sums and the differences of cubes. Here are the two formulas:. You'll learn in more advanced classes how they came up with these formulas.

For now, just memorize them. To help with the memorization, first notice that the terms in each of the two factorization formulas are exactly the same. Then notice that each formula has only one "minus" sign. The distinction between the two formulas is in the location of that one "minus" sign:. Some people use the mnemonic " S O A P " to help keep track of the signs; the letters stand for the linear factor having the "same" sign as the sign in the middle of the original expression, then the quadratic factor starting with the "opposite" sign from what was in the original expression, and finally the second sign inside the quadratic factor is "always positive".

Whatever method best helps you keep these formulas straight, use it, because you should not assume that you'll be given these formulas on the test. You should expect to need to know them. Note: The quadratic portion of each cube formula does not factor , so don't waste time attempting to factor it. These sum- and difference-of-cubes formulas' quadratic terms do not have that " 2 ", and thus cannot factor. When you're given a pair of cubes to factor, carefully apply the appropriate rule.

By "carefully", I mean "using parentheses to keep track of everything, especially the negative signs". Here are some typical problems:.

This is equivalent to x 3 — 2 3. With the "minus" sign in the middle, this is a difference of cubes. To do the factoring, I'll be plugging x and 2 into the difference-of-cubes formula. Doing so, I get:. The first term contains the cube of 3 and the cube of x.

But what about the second term? Before panicking about the lack of an apparent cube, I remember that 1 can be regarded as having been raised to any power I like, since 1 to any power is still just 1.

In this case the power I'd like is 3 , since this will give me a sum of cubes. This means that the expression they've given me can be expressed as:. So, to factor, I'll be plugging 3 x and 1 into the sum-of-cubes formula.

This gives me:. First, I note that they've given me a binomial a two-term polynomial and that the power on the x in the first term is 3 so, even if I weren't working in the "sums and differences of cubes" section of my textbook, I'd be on notice that maybe I should be thinking in terms of those formulas. Looking at the other variable, I note that a power of 6 is the cube of a power of 2 , so the other variable in the first term can be expressed in terms of cubing, too; namely, as the cube of the square of y.

The second term is 64 , which I remember is the cube of 4. If I didn't remember, or if I hadn't been certain, I'd have grabbed my calculator and tried cubing stuff until I got the right value, or else I'd have taken the cube root of So I now know that, with the "minus" in the middle, this is a difference of two cubes; namely, this is:. I know that 16 is not a cube of anything; it's actually equal to 2 4. What's up? What's up is that they expect me to use what I've learned about simple factoring to first convert this to a difference of cubes.

I can get 8 from 16 by dividing by 2. What happens if I divide by 2? I get , which is the cube of 5. So what they've given me can be restated as:. But what about that "minus" sign in front? Since neither of the factoring formulas they've given me includes a "minus" in front, maybe I can factor the "minus" out? Now what's inside the parentheses is a sum of cubes, which I can factor.

I've got the sum of the cube of x and the cube of 5 , so:. You can use the Mathway widget below to practice factoring a sum of cubes.

Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's. Or skip the widget and continue with the lesson. Please accept "preferences" cookies in order to enable this widget. Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade. Page 1 Page 2 Page 3 Page 4. All right reserved. Web Design by. Skip to main content.

Sums and Differences of Cubes Diff. Purplemath The other two special factoring formulas you'll need to memorize are very similar to one another; they're the formulas for factoring the sums and the differences of cubes.

Content Continues Below. So I now know that, with the "minus" in the middle, this is a difference of two cubes; namely, this is: xy 2 3 — 4 3. Share This Page. Terms of Use Privacy Contact.

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