Expansion-Square Of Binomial –
The product of an algebraic expression multiplied by itself (any number of times) when expressed as a polynomial is called an expansion. In fact, the term ‘Expansion’ is used for the process of such multiplication. In this section, we will look at some expansions and their corollaries. A corollary is a result or relation that follows from another result or relation that is known.
Square of Binomials –
Let, a & b be two terms, we will find the expansions of the squares of the sum of these two terms. In other words, we will find the expansions of (a+b)² = a² + 2ab + b²
Proof - (a+b)² = (a+b) (a+b)
= a(a+b) + b (a+b)
= a²+ ab + ba + b²
= a²+ 2ab + b²
So, (a+b)² = a²+ 2ab + b² (proven)
We can express these as follows –
(sum of two terms)² = (first term)² + 2 X (first term) X (second term) + (second term)²
Corollaries –
Two corollaries follow from the expansion we have just discussed.
1) (a+b)² - 2ab = a² + 2ab + b² -2ab = a² + b²
So, we can express a²+ b² = (a+b)²- 2ab
2) (a+b)²- (a²+b²) = a²+ 2ab + b²- a²- b² = 2ab
So, 2ab = (a+b)²- (a²+b²)
Let, a & b be two terms, we will find the expansions of the squares of the difference of these two terms. In other words, we will find the expansions of (a-b)² = a² - 2ab + b²
Proof - (a-b)² = (a-b) (a-b)
= a(a-b) - b(a-b)
= a²- ab - ba + b²
= a²- 2ab + b²
So, (a-b)² = a²- 2ab + b² (proven)
We can express these as follows –
(sum of two terms)² = (first term)² - 2 X (first term) X (second term) + (second term)²
Corollaries –
Several corollaries follow from the two expansions we have discussed.
A) (a-b)² +2ab = a² - 2ab + b² + 2ab = a² + b²
So, a²+ b² = (a-b)²+ 2ab
B) a² + b² - (a-b)² = a² + b² - (a²-2ab+b²)
= a²+ b²- a²+ 2ab - b²
= 2ab
So, 2ab = a² + b²- (a-b)²
C) (a - b)² + 4ab = a²- 2ab+ b²+ 4ab = a² + 2ab + b² = (a + b)²
So, (a + b)² = (a – b)² + 4ab
D) (a+b)² - 4ab = a²+ 2ab+ b²- 4ab = a² - 2ab + b² = (a - b)²
So, (a – b)² = (a + b)²- 4ab
1 1
E) ------ [(a + b)² - (a – b)²] = ------ [ a²+ 2ab+ b²- (a²- 2ab+ b²)]
4 4
1 1
= ------ [a²+ 2ab+ b² - a² +2ab - b² ] = ------- X 4ab = ab
4 4
1
So, ab = -------- [(a + b)² - (a – b)²]
4
1 1
F) ------ [(a + b)² + (a – b)²] = ------ [ a²+ 2ab + b² + a²- 2ab + b²]
2 2
1 1
= ------- (a² + b² + a² + b²) = ------- (2a² + 2b²)
2 2
1
= ------- X 2 (a² + b²) = (a²+ b²)
2
1
So, a² + b² = ------ [(a + b)² + (a – b)²]
2