'Binomial' is a word meaning 'two terms', and is used in algebra to mean expressions such as a + 2 and 2x − y. (Compare with the word 'polynomial' - 'an expression of more than two algebraic terms'.)
You are probably familiar with expressions such as
| (a + b)2 | = (a + b) (a + b) |
| = a2 + ab + ba + b2 | |
| = a2 + 2ab + b2 |
Worked Examples
Here are six statements about binomials. Which of these are true, and which false?
2(a + b) = 2a + 2b
Show me| 2(a + b) | = 2 × a + 2 × b | |
| = 2a + 2b | so this statement is true. |
(a + b) ÷ 2 = (a ÷ 2) + (b ÷ 2)
Show me (a + b) ÷ 2 = a ÷ 2 + b ÷ 2 so this statement is true.(a + b)2 = a2 + b2
Show me| (a + b)2 | = a2 + 2ab + b2 |
| ≠ a2 + b2 (unless a or b equals 0) so statement (c) is false. |
(a + b)3 = a3 + b3
Show me| (a + b)^3 | = (a + b)(a + b)^2 |
| = (a + b)(a^2 + 2ab + b^2) | |
| = (a^3 + 2a^2b + ab^2) + (ba^2 + 2ab^2 + b^3) | |
| = a^3 + 3a^2b + 3ab^2 + b^3 so this statement, too, is false (unless a or b equals zero). |
\sqrt{(a + b)} = \sqrt{a} + \sqrt{b}
Show me\sqrt{a+b} ≠ \sqrt{a} + \sqrt{b}
as (\sqrt{a + b})^2 = a + b
but (\sqrt{a} + \sqrt{b}) = (\sqrt{a})^2 + 2\sqrt{a}\sqrt{b} + (\sqrt{b})^2
= a + 2\sqrt{ab} + b so this statement is false (unless a or b equals zero).
2^{(a + b)} = 2^a + 2^b
Show meThis statement is false as can easily be shown by substituting a = b = 2.
Then 2^{a + b} = 2^{2 + 2} = 2^4 = 16
and 2^a + 2^b = 2^2 + 2^2 = 4 + 4 = 8 So statement (f) is not true in general.
Note that:
(a + b)^2 = a^2 + 2ab + b^2
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
These are simple examples of binomial expansions.
Expand (1 + x)4.
| (1 + x)^4 | = (1 + x)^2 (1 + x)^2 |
| = (1 + 2x + x^2)(1 + 2x + x^2) | |
| = 1(1 + 2x + x^2) + 2x(1 + 2x + x^2) + x^2(1 + 2x + x^2) | |
| = 1 + 2x + x^2 + 2x + 4x^2 + 2x^3 + x^2 + 2x^3 + x^4 | |
| = 1 + 4x + 6x^2 + 4x^3 + x^4 |
Exercises
