A binomial is an algebraic expression which has exactly two
terms, for example, a ± b. Its power is indicated by a superscript. For
example, (a ± b)^{2} is a power of the binomial a ± b, the index being 2.

A trinomial is an algebraic expression which has exactly
three terms, for example, a ± b ± c. Its power is also indicated by a
superscript. For example, (a ± b ± c)^{3} is a power of the trinomial a ± b ± c,
whose index is 3.

**Expansion of (a ± b) ^{2}**

(a +b)\(^{2}\)

= (a + b)(a + b)

= a(a + b) + b(a+ b)

= a\(^{2}\) + ab + ab + b\(^{2}\)

= a\(^{2}\) + 2ab + b\(^{2}\).

(a - b)\(^{2}\)

= (a - b)(a - b)

= a(a - b) - b(a - b)

= a\(^{2}\) - ab - ab + b\(^{2}\)

= a\(^{2}\) - 2ab + b\(^{2}\).

Therefore, (a + b)\(^{2}\) + (a - b)\(^{2}\)

= a\(^{2}\) + 2ab + b\(^{2}\) + a\(^{2}\) - 2ab + b\(^{2}\)

= 2(a\(^{2}\) + b\(^{2}\)), and

(a + b)\(^{2}\) - (a - b)\(^{2}\)

= a\(^{2}\) + 2ab + b\(^{2}\) - {a\(^{2}\) - 2ab + b\(^{2}\)}

= a\(^{2}\) + 2ab + b\(^{2}\) - a\(^{2}\) + 2ab - b\(^{2}\)

= 4ab.

**Corollaries:**

(i) (a + b)\(^{2}\) - 2ab = a\(^{2}\) + b\(^{2}\)

(ii) (a - b)\(^{2}\) + 2ab = a\(^{2}\) + b\(^{2}\)

(iii) (a + b)\(^{2}\) - (a\(^{2}\) + b\(^{2}\)) = 2ab

(iv) a\(^{2}\) + b\(^{2}\) - (a - b)\(^{2}\) = 2ab

(v) (a - b)\(^{2}\) = (a + b)\(^{2}\) - 4ab

(vi) (a + b)\(^{2}\) = (a - b)\(^{2}\) + 4ab

(vii) (a + \(\frac{1}{a}\))\(^{2}\) = a\(^{2}\) + 2a ∙ \(\frac{1}{a}\) + (\(\frac{1}{a}\))\(^{2}\) = a\(^{2}\) + \(\frac{1}{a^{2}}\) + 2

(viii) (a - \(\frac{1}{a}\))\(^{2}\) = a\(^{2}\) - 2a ∙ \(\frac{1}{a}\) + (\(\frac{1}{a}\))\(^{2}\) = a\(^{2}\) + \(\frac{1}{a^{2}}\) - 2

Thus, we have

**1. (a +b)\(^{2}\) = a\(^{2}\) + 2ab + b\(^{2}\).**

**2. (a - b)\(^{2}\) = a\(^{2}\) - 2ab + b\(^{2}\).**

**3. (a + b)\(^{2}\) + (a - b)\(^{2}\) = 2(a\(^{2}\) + b\(^{2}\))**

**4. (a + b)\(^{2}\) - (a - b)\(^{2}\) = 4ab.**

**5. (a + \(\frac{1}{a}\))\(^{2}\) = a\(^{2}\) + \(\frac{1}{a^{2}}\) + 2**

**6. (a - \(\frac{1}{a}\))\(^{2}\) = a\(^{2}\) + \(\frac{1}{a^{2}}\) - 2**

Solved Example on Expansion of (a ± b)^{2}

1. Expand (2a + 5b)^2.

Solution:

(2a + 5b)^2

= (2a)^2 + 2 ∙ 2a ∙ 5b + (5b)^2

= 4a^2 + 20ab + 25b^2

2. Expand (3m - n)^2.

Solution:

(3m - n)^2.

= (3m)^2 - 2 ∙ 3m ∙ n + n^2

= 9m^2 - 6mn + n^2

3. Expand (2p + \(\frac{1}{2p}\))^2

Solution:

(2p + \(\frac{1}{2p}\))^2

= (2p)^2 + 2 ∙ 2p ∙ \(\frac{1}{2p}\) + (\(\frac{1}{2p}\))^2

= 4p^2 + 2 + \(\frac{1}{4p^{2}}\)

4. Expand (a - \(\frac{1}{3a}\))^2

Solution:

(a - \(\frac{1}{3a}\))^2

= a^2 - 2 ∙ a ∙ \(\frac{1}{3a}\) + (\(\frac{1}{3a}\))^2

= a^2 - \(\frac{2}{3}\) + \(\frac{1}{9a^{2}}\)

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