Factorising two term cubic expressions.
Steps to follow.
Cubic expressions of the form x3 + y3 and x3 - y3 can be factorised easily by adopting a straight forward model:
1. The sum of 2 cubes - x3 + y3 - becomes (x + y)(x2 - xy + y2). To accomplish this factorisation:
| Step 1: | Write the two terms as cubes. | Write x3 as (x)3.
Write 27y3 as (3y)3. |
| Step 2: | Form the first factorised term by adding the two terms in brackets. | x3 + y3 = (x)3 + (y)3 has its first term as (x + y) 8x3 + 27y3 = (2x)3 + (3y)3 has its first term as (2x + 3y) Hence the sign used here is the same as that used in the original expression. |
| Step 3: | Form the second factorised term as follows: | (2x)3 + (3y)3 becomes |
| Write the square of the first term. | (2x)2 becomes (4x2 | |
| SUBTRACT the product of the two bracketed terms. | (4x2 - (2x)(3y) becomes (4x2 - 6xy Hence the sign used for the 2nd term is the opposite to that used in the original expression. |
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| Add the square of the second term | (4x2 - 6xy + (3y)2) = (4x2 - 6xy + 9y2) | |
| Writing the combined terms: | (2x + 3y)(4x2 - 6xy + 9y2) |
2. The DIFFERENCE between 2 cubes - x3 - y3 - becomes (x - y)(x2 + xy + y2). To accomplish this factorisation:
| Step 1: | Write the two terms as cubes. | Write x3 as (x)3.
Write 27y3 as (3y)3. |
| Step 2: | Form the first factorised term by subtracting the two terms in brackets. | x3- y3 = (x)3 - (y)3 has its first term as (x - y) 8x3 - 27y3 = (2x)3 - (3y)3 has its first term as (2x - 3y) Hence the sign used for the second term is the same as that |
| Step 3: | Form the second factorised term as follows: | (2x)3 + (3y)3 becomes |
| Write the square of the first term. | (2x)2 becomes (4x2 | |
| ADD the product of the two bracketed terms. | (4x2 + (2x)(3y) becomes (4x2 + 6xy Hence the sign used for the 2nd term is the opposite to that used in the original expression. |
|
| Add the square of the second term | (4x2 + 6xy + (3y)2) = (4x2 + 6xy + 9y2) | |
| Writing the combined terms: | (2x - 3y)(4x2 + 6xy + 9y2) |
Sometimes it is possible to take out a common factor from the two terms before beginning the factorisation as described. This is the parallel approach used when factorising some quadratic expressions. Hence a good strategy as a first step is to check if a common factor can be identified.