Lesson The cube of the sum formula

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<H2>The cube of the sum formula</H2>

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

This formula is

{{{(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3}}}.

The formula is valid for any real numbers {{{a}}} and {{{b}}}. 
It is valid for the complex numbers too.

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

{{{(a + b)^3 = (a + b)*(a + b)^2 = (a + b)*(a^2 + 2ab + b^2) = a*(a^2 + 2ab + b^2) + b*(a^2 + 2ab + b^2) = a^3 + 2a^2*b + a*b^2 + a^2*b + 2a*b^2 + b^3 = a^3 + 3a^2b + 3ab^2 + b^3}}}.

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.<TABLE cellspacing="20"> 
 <TR>
 <TD>
{{{drawing( 400, 350, -2.5, 5.5, -1.5, 5.5,

line   ( 0,  0,  5,  0),
            line   ( 5,  0,  5,  5),
            line   ( 0,  5,  5,  5),
            line   ( 0,  0,  0,  5),

green(line   ( 0,  0, -2, -1)),
      green(line   ( 5,  0,  3, -1)),
      green(line   ( 0,  5, -2,  4)),
      green(line   ( 5,  5,  3,  4)),

line   (-2, -1,  3, -1),
            line   ( 3, -1,  3,  4),
            line   (-2,  4,  3,  4),
            line   (-2, -1, -2,  4),

red(line   ( 0,    0,    3.5,  0)),
        red(line   ( 3.5,  0,    3.5,  3.5)),
        red(line   ( 0,    3.5,  3.5,  3.5)),
        red(line   ( 0,    0,    0,    3.5)),

red(line   ( 0,    0,   -1.4, -0.7)),
        red(line   ( 3.5,  0,    2.1, -0.7)),
        red(line   ( 0,    3.5, -1.4,  2.8)),
        red(line   ( 3.5,  3.5,  2.1,  2.8)),

red(line   (-1.4, -0.7,  2.1, -0.7)),
        red(line   ( 2.1, -0.7,  2.1,  2.8)),
        red(line   (-1.4,  2.8,  2.1,  2.8)),
        red(line   (-1.4, -0.7, -1.4,  2.8)),

blue(line   ( 2.1,  2.8,  2.1,  4.3)),
       blue(line   ( 2.1,  2.8,  3.6,  2.8)),
       blue(line   ( 2.1,  2.8,  1.5,  2.5)),

blue(line   ( 2.1,  4.3,  3.6,  4.3)),
       blue(line   ( 3.6,  4.3,  3.6,  2.8)),

blue(line   ( 1.5,  2.5,  3.0,  2.5)),
       blue(line   ( 3.0,  2.5,  3.6,  2.8)),

blue(line   ( 2.1,  4.3,  1.5,  4.0)),
       blue(line   ( 1.5,  2.5,  1.5,  4.0)),

locate ( 1.6,   0.3,  a),
            locate ( 0.6,  -0.4,  a),
            locate (-0.40, -0.15, a),
            locate ( 2.55, -0.15, a),

locate (-1.35,  1.2,  a),
            locate ( 0.05,  1.8,  a),
            locate (-0.40,  3.35, a),

locate ( 1.87,  1.2,  a),
            locate ( 3.3,   1.8,  a),

locate ( 0.6,   3.1,  a),

locate ( 4.2,   0.3,  b),
            locate ( 0.05,  4.4,  b),
            locate (-1.80, -0.6,  b),

locate ( 0.5,  -1.0,  a+b),
            locate ( 2.0,   5.3,  a+b),
            locate ( 4.0,  -0.5,  a+b),
            locate (-1.8,   4.8,  a+b),

locate ( 3.65, 3.7, b),
 locate ( 2.30, 2.5, b),
 locate ( 1.55, 4.4, b)
)}}}
Figure. Illustration to the cube of the sum formula 
 </TD>
 <TD>
The Figure shows the big cube with the side length {{{a+b}}}. 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 {{{a}}} and {{{b}}}. The cube with the side {{{a}}} is shown by red lines. The cube with the side {{{b}}} is shown by blue lines. The volumes of these smaller cubes are {{{a^3}}} and {{{b^3}}} respectively.

Now, let us consider the cube with the side measure {{{a}}}. It has three internal sides (surfaces) that are squares with the side measure {{{a}}}. 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 {{{a}}}. The third dimension of these parallelepipeds is equal to {{{b}}}, so the volume of each of the three parallelepiped is equal to {{{a^2*b}}}.

Next, consider the cube with the side measure {{{b}}}. It has its own three internal sides (surfaces) that are squares with the side measure {{{b}}}. 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 {{{b}}}. The third dimension of these parallelepipeds is equal to {{{a}}}, so the volume of each of the three parallelepiped is equal to {{{a*b^2}}}. 
 </TD>
 </TR>
</TABLE> 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

{{{(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3}}}.

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,

{{{(cx^2 + dx)^3 = c^3*x^6 + 3c^2*d*x^5 + 3c*d^2*x^4 + d^3*x^3}}}.

Here {{{cx^2}}} and {{{dx}}} are monomials that you can treat like the symbols {{{a}}} and {{{b}}} 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.

<H3>Example 1</H3>Simplify the expression {{{x^3 + 6x^2 + 12x + 8}}}.

Solution

Apply the cube of the sum formula. You have

{{{x^3 + 6x^2 + 12x + 8 = x^3 + 3*2*x^2 + 3*2^2*x + 8 = (x + 2)^3}}}.

<H3>Example 2</H3>Simplify the expression {{{27a^3 + 27a^2*b + 9ab^2 + b^3}}}.

Solution

Apply the cube of the sum formula. You get

{{{27a^3 + 27*a^2 + 9ab^2 + b^3 = (3a)^3 + 3*(3a)^2*b + 3*(3a)*b^2 + b^3 = (3a + b)^3}}}.

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

<H3>Example 3</H3>Factor the polynomial {{{27x^6 + 27x^4 + 9x^2 + 1}}}.

Solution

Apply the cube of the sum formula. You get

{{{27x^6 + 27x^4 + 9x^2 + 1 = (3x^2)^3 + 3*(3x^2)^2 + 3*(3x^2) + 1 = (3x^2 + 1)^3}}}.

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

<H3>Example 4</H3>Factor the polynomial {{{27x^7 + 27x^5 + 9x^3 + x}}}.

Solution

First take the common factor {{{x}}} out the brackets. You get

{{{27x^7 + 27x^5 + 9x^3 + x = x*(27x^6 + 27x^4 + 9x^2 + 1)}}}.

Now, factor the polynomial {{{27x^6 + 27x^4 + 9x^2 + 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^7 + 27x^5 + 9x^3 + x = x*(3x^2 + 1)^3}}}.

<H3>Summary</H3>The cube of the sum formula

{{{(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3}}}

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

For similar lesson see

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/The_cube_of_a_difference.lesson>The cube of the difference formula</A>

under the current topic in this site.

For the list of all shortcut cubic multiplication formulas see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Review-of-shortcut-cubic-multiplication-formulas.lesson>OVERVIEW of shortcut cubic multiplication formulas</A>

under the current topic in this site.

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-I.