Lesson The cube of the sum formula

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The cube of the sum formula

This lesson is focused on the useful formula of the cube of the sum of two numbers.
This formula is

%28a+%2B+b%29%5E3+=+a%5E3+%2B+3a%5E2b+%2B+3ab%5E2+%2B+b%5E3

.

The formula is valid for any real numbers

and

.
It is valid for the complex numbers too.

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

.

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 square of a sum formula has a remarkable geometric illustration.
It is presented in the Figure below.

Figure. Illustration to the cube of the sum formula

The Figure shows the big cube with the side length a%2Bb

. Its edges are shown by black and green segments. The cube is subdivided into parts. Two of these parts are smaller cubes with the side sizes

and

. The cube with the side

is shown by red lines. The cube with the side

is shown by blue lines. The volumes of these smaller cubes are a%5E3

and

respectively.

Now, let us consider the cube with the side measure

. It has three internal sides (surfaces) that are squares with the side measure

. Each of these three internal surfaces is common to the cube and one of the three parallelepipeds that are adjacent to the cube. Thus the base of each of these three parallelepipeds is the square with the side measure

. The third dimension of these parallelepipeds is equal to

, so the volume of each of the three parallelepiped is equal to a%5E2%2Ab

.

Next, consider the cube with the side measure

. It has its own three internal sides (surfaces) that are squares with the side measure

. Again, each of these three internal surfaces is common to the cube and one of the three parallelepipeds that are adjacent to the cube. Thus the base of each of these three parallelepipeds is the square with the side measure

. The third dimension of these parallelepipeds is equal to

, so the volume of each of the three parallelepiped is equal to a%2Ab%5E2

Since the volume of the large cube is the sum of the volumes of its parts, the Figure illustrates the cube of the sum formula

%28a+%2B+b%29%5E3+=+a%5E3+%2B+3a%5E2b+%2B+3ab%5E2+%2B+b%5E3

.

You should memorize this formula. The illustration could help you memorize it.

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

.

Here cx%5E2

and

are monomials that you can treat like the symbols

and

in the cube of the sum formula.
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 cube of the sum formula is simply the useful shortcut formula.
It may help you in different ways when you need to simplify the polynomial expressions or to factor polynomials.

Example 1

Simplify the expression x%5E3+%2B+6x%5E2+%2B+12x+%2B+8

.

Solution
Apply the cube of the sum formula. You have

Example 2

Simplify the expression 27a%5E3+%2B+27a%5E2%2Ab+%2B+9ab%5E2+%2B+b%5E3

.

Solution
Apply the cube of the sum formula. You get

.

Note. Pay attention how the terms are grouped to follow the pattern of the cube of the sum formula.

Example 3

Factor the polynomial 27x%5E6+%2B+27x%5E4+%2B+9x%5E2+%2B+1

.

Solution
Apply the cube of the sum formula. You get

.

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

Example 4

Factor the polynomial 27x%5E7+%2B+27x%5E5+%2B+9x%5E3+%2B+x

.

Solution
First take the common factor

out the brackets. You get

27x%5E7+%2B+27x%5E5+%2B+9x%5E3+%2B+x+=+x%2A%2827x%5E6+%2B+27x%5E4+%2B+9x%5E2+%2B+1%29

.

Now, factor the polynomial 27x%5E6+%2B+27x%5E4+%2B+9x%5E2+%2B+1

, which is in the brackets on the right side, as it is done in the Example 3 above.
Finally, you get the required factorization

27x%5E7+%2B+27x%5E5+%2B+9x%5E3+%2B+x+=+x%2A%283x%5E2+%2B+1%29%5E3

27x%5E7+%2B+27x%5E5+%2B+9x%5E3+%2B+x+=+x%2A%283x%5E2+%2B+1%29%5E3

Summary

The cube of the sum formula

%28a+%2B+b%29%5E3+=+a%5E3+%2B+3a%5E2b+%2B+3ab%5E2+%2B+b%5E3

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

For similar lesson see
The cube of the difference formula
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.

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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