The sum of two cubes is the 7th case of factoring algebraic expressions, its reasoning is the same as in sum of two cubes, reasoning that clarifies how and when we should use it, observe the demonstration below:
Given any two numbers x and y. If we subtract we will get: x – y, if we build an algebraic expression with the two numbers we will get: x2 + xy + y2, thus, we must multiply the two expressions found.
(x - y) (x2 + xy + y2) it is necessary to use the distributive property;
x3 + x2y + xy2 - x2y –xy2 -y3 join similar terms;
x3 -y3 is an algebraic expression of two terms, the two are cubed and subtracted.
Thus, we can conclude that x3 -y3 is a general form of the sum of two cubes where
x and y can take any real value.
The factored form of x3 -y3 will be (x - y) (x2 + xy + y2).
See some examples:
Example1
If we have to factor the following 8x algebraic expression3 – 27, we should note that it has two terms. Remembering the factoring cases, the only case that factors two terms is the difference of two squares, sum of two cubes and the difference of two cubes.
In the example above the two terms are cubed and between them there is a subtraction, so we should use the 7th case of factorization (difference of two cubes), to factorize we must write the algebraic expression 8x3 – 27 as follows:
(x - y) (x2 + xy + y2). By taking the cubic roots of the two terms, we have: 8x3 – 27
The 8x Cubic Root3 is 2x and the cubic root of 27 is 3. Now, just substitute values, instead of x we'll put 2x and instead of y we'll put 3 in factored form
(x - y) (x2 + xy + y2), looking like this:
(2x - 3) ((2x)2 + 2x. 3 + 32)
(2x - 3) (4x2 + 6x + 9)
So (2x - 3) (4x2 + 6x + 9) is the factored form of the 8x algebraic expression3 – 27.
Example 2
To solve the factorization using the difference of two cubes we must follow the same steps as in the previous example. Factoring the algebraic expression r3 – 64 we have: The cubic roots of r3 is r and 64 is 4, substituting r for x and r for y for 4.
(r – 4) (r2 + 4r + 16) is the factored form of r3 – 64.
by Danielle de Miranda
Graduated in Mathematics
Brazil School Team
Algebraic Expression Factorization
Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/diferenca-dois-cubos.htm
