Difference of squares is a special type of binomial in algebra. Care should be taken while factoring polynomials that students recognize as the difference of squares binomial.
Generally, a binomial of the type “a² – b²” is called a difference of squares as it is apparent from its terms, “a²” and “b²” and a negative sign between them. To factor difference of squares, there is a special rule to memorize and if student memorize it in early grades such as grade 8, then it is very very useful in solving many of the algebra problem.
The factors of the general binomial “a² – b²” are as shown below:
a² – b² = (a + b) (a – b) or some students can write it by switching the brackets having the bracket with negative sign first; a² – b² = (a – b) (a + b), which makes no difference.
To memorize this; keep in mind to get rid of the square and making two brackets with opposite signs that is one bracket with positive sign between the terms and the other with the negative sign between the terms.
Note that, both brackets should have different signs and it doesn’t matter which sign is in the first bracket and which is in the second.
Let’s do the following examples to understand the difference of squares further:
1. a² – b² = (a + b) (a – b)
2. p² – q² = (p +q) (p -q)
3. x² – y² = (x + y) (x – y)
The above examples show that by changing the variables, the procedure to factor difference of squares doesn’t change. Also, I chose to write the bracket with “plus sign” first in my factors. You can write the bracket with negative sign first, which is right too.
4. 9a² – b²
This is the example I want to explain further, first term is “9a²” where 9 don’t have a square, but to factor difference of squares, both the terms should be perfect squares. But if you have knowledge of perfect squares, 9 can be written as 3² and hence “9a²” can be written as “(3a)²” to complete the square of the first term.
Let’s rewrite both the terms with squares as shown below:
(3a)² – b², now compare it with the general binomial, “a² – b²” we have “a” replaced by “3a”. Hence, replace “a” by “3a” in the factors too.
(3a + b) (3a – b) are the factors for “9a² – b²”
Rewrite all the steps together in a way to show your work:
9a² – b²
= (3a)² – b²
= (3a – b) (3a + b)
5. 16x² – 25y²
= (4x)² – (5y)²
= (4x – 5y) (4x +5y)
That’s all about the difference of squares method for factoring these special binomials.