Lesson The cube of the difference formula

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

Probably, you just know the cube of the sum formula

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

(see the lesson <A HREF=http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/The_cube_of_a_sum.lesson>The cube of the sum formula</A> under the current module in this site).
In THIS lesson you will learn about the close formula for the cube of the difference

{{{(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.

You can obtain the formula for the cube of a difference from the cube of the sum formula by substituting there the opposite number {{{-b}}} instead of {{{b}}}.
Or, you can derive it independently, simply performing straightforward calculation:

{{{(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.

You should memorize this formula.

The cube of the difference 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 difference 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 difference 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 difference 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 difference 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 difference formula.

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

Solution

Apply the cube of the difference 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 difference 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 difference 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_sum_formula.lesson>The cube of the sum 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.