Sum Cube and Difference Cube are two types of notable products, where two terms are being added or subtracted and then cubed, that is, with an exponent equal to 3.
(x + y) ³ -> sum cube
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(x – y) ³ -> cube of difference
The sum cube can also be written as (x+y). (x+y). (x + y) and the cube of the difference as (x – y). (x – y). (x – y).
These products receive the name of notable products for the importance they have, since they appear frequently in algebraic calculations.
Now, remember that, in mathematics, the same expression can be written in another way, but without changing its value. For example, x + 1 + 1 can be written simply as x + 2.
Often, when we rewrite an expression, we can simplify and solve many algebraic problems. Therefore, let's see another way of writing the cube of the sum and the cube of the difference, developing them algebraically.
sum cube
O sum cube is the remarkable product (x + y) ³, which is the same as (x + y). (x+y). (x+y). In this way, we can write:
(x + y) ³ = (x + y). (x+y). (x + y)
Now, considering that (x + y). (x + y) = (x + y) ² = x² + 2xy + y², the cube of the sum can be written as:
(x + y) ³ = (x + y). (x² + 2xy + y²)
Multiplying the polynomial (x + y) by (x² + 2xy + y²), we can see that:
(x + y) ³ = x³ + 2x²y + xy² + x²y + 2xy² + y³
Adding like terms, we have that the cube of the sum is given by:
(x + y) ³ = x³ + 3x²y + 3xy² + y³
Example:
Develop each cube algebraically:
a) (x + 5)²
(x + 5)² = (x) ³ + 3.(x) ².(5) + 3.(x).(5)² + (5)³
= x³ + 3.x².5 + 3.x.25 + 125
= x³ +15x² +75x + 125
b) (1 + 2b) ³
(1 + 2b) ³ = (1)³ + 3.(1)².(2b) + 3.(1).(2b) ² + (2b) ³
= 1 + 3.1.2b + 3.1.4b² + 8b³
= 1 + 6b + 12b² + 8b³
difference cube
O difference cube is the notable product (x – y) ³, which is the same as (x – y). (x – y). (x – y). So, we have to:
(x – y) ³ = (x – y). (x – y). (x – y)
Like (x – y). (x – y) = (x – y) ² = x² – 2xy + y², the cube of the difference can be written as:
(x – y) ³ = (x – y). (x² – 2xy + y²)
Multiplying (x – y) by (x² – 2xy + y²), we can see that:
(x – y) ³ = x³ – 2x²y + xy² – x²y + 2xy² – y³
Adding like terms, we have that the cube of the difference is given by:
(x – y) ³ = x³ – 3x²y + 3xy² – y³
Example:
Develop each cube algebraically:
a) (x – 2)³
(x – 2)³ = (x) ³ – 3.(x) ².(2) + 3.(x).(2)² – (2)³
= x³ – 3.x².2 + 3.x.4 – 8
= x³ – 6x² + 12x – 8
b) (2a – b) ³
(2a – b) ³ = (2a) ³ – 3.(2a) ².(b) + 3.(2a).(b²) – (b) ³
= 8a³ – 3.4a².b + 3.2a.b² – b³
= 8a³ – 12a²b + 6ab² – b³
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