# CLASS-8ALGEBRAIC EXPANSION - SQUARE OF BINOMIAL

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