Lesson The sum of cubes formula

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The sum of cubes formula


The sum of cubes formula is

a%5E3+%2B+b%5E3+=+%28a%2Bb%29%2A%28a%5E2+-+ab+%2B+b%5E2%29.

The formula is valid for any real numbers a and b.

The proof of the formula is very simple. It follows straightforward from the direct calculations:


                                                                                    +      a%5E2%2Ab+-+a%2Ab%5E2+%2B+b%5E3 = a%5E3+%2B+b%5E3.

As you see, the distributive and the commutative properties of addition and multiplication operations over the real numbers are used in derivation the formula.

The sum of cubes formula is applicable not only to numbers. It is applicable for binomials too. For example,

x%5E3+%2B+1+=+%28x+%2B+1%29%2A%28x+-+x+%2B+1%29.

You can check validity of the last formula directly by performing all relevant calculations: opening the brackets, multiplying the terms and combining the like terms.
You will get the same result. It is not surprising, because addition and multiplication operations over polynomials have the same distributive and commutative properties
as over the real numbers. Thus, the sum of cubes formula is simply the useful shortcut formula.
It may help you when you need to factor polynomials.

Example 1

Factor the binomial 8x%5E3+%2B+27.

Solution

Apply the sum of cubes formula. You have
.


Example 2

Factor the binomial 125x%5E3+%2B+64.

Solution

Apply the sum of cubes formula. You have
.

Note. Pay attention how the brackets are used to group monomials according the pattern of the sum of cubes formula.


Example 3

Factor the binomial x%5E6+%2B+1.

Solution

Note that x%5E6+%2B+1+=+%28%28x%5E2%29%5E3+%2B+1%29. Now, apply the sum of cubes formula:
.


Example 4

Factor the expression a%5E6+%2B+b%5E6.

Solution
Note that a%5E6+%2B+b%5E6+=+%28%28a%5E2%29%5E3+%2B+%28b%5E2%29%5E3%29. Now, apply the sum of cubes formula:
.


Summary

The sum of cubes formula

a%5E3+%2B+b%5E3+=+%28a+%2B+b%29%2A%28a%5E2+-+ab+%2B+b%5E2%29

is useful shortcut multiplication formula.
At the same time, you can use it to factor binomials when applicable.


For similar lessons see
    The cube of the sum formula and
    The cube of the difference formula
    The difference of cubes formula
under the current topic in this site.

The sum of cubes formula is often used in rationalizing fractions by making their denominator free of cubic roots. For details and examples see the lesson
    HOW TO rationalize a fraction by making its denominator free of cubic roots
under the current topic in this site. It is used also in simplifying rational expressions. For details and examples see the lesson
    Simplifying rational expressions with the use the shortcut multiplication formulas
under the current topic in this site.

For the list of all shortcut cubic multiplication formulas see the lesson
    OVERVIEW of shortcut cubic multiplication formulas
under the current topic in this site.


For factoring the binomials of high degrees  x%5En-a%5En,   x%5En%2Ba%5En  see the lessons
    Factoring the binomials x%5En-a%5En,
    Factoring the binomials x%5En%2Ba%5En for odd degrees,
    OVERVIEW of Factoring the binomials x%5En-a%5En and x%5En%2Ba%5En
under the current topic in this site.

Use this file/link  ALGEBRA-I - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-I.

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