Operations on Binomials

Definition of binomial:
We define binomial as follows: An algebraic expression consisting of constants and variables that contains only two terms is called a binomial.

Binomial examples:
For example: 2x + 3, 4x + 5y, x^2 + 1, 3x^2 + 6x. Each of those have exactly two terms and hence are called binomials.

Operations on binomials:

1. Adding binomials and subtracting binomials:

For addition or subtraction of two or more binomials,
a) We collect the like terms together.
b) Find the sum or difference of the numerical coefficients of these terms.
c) The resulting expression should be in the simplest form and can be written according to the ascending or descending order of terms.

For example 1: Add 7a^3 + 3a^2b , 2a^3 ? 4a^2b, a^3 ? a^2b
Solution:
Step 1: (7a^3 + 2a^3 + a^3) + (3a^2b ? 4a^2b ? a^2b)
Step 2: (10a^3) + (-2a^2b)
Step 3: 10a^3 ? 2a^2b

2.

Multiply binomials:

For multiplying binomials we use the FOIL method. F is for First, O is for Outer, I is inner and L is Last. When we are to multiply two binomials, they would look like this: (a + b)*(c + d). So we have two brackets. The first terms of the two? brackets are multiplied first, so F = a*c = ac. The outer terms of the two brackets are multiplied next, so O = a*d = ad. Then the inner terms of both the brackets are multipled, so I = bc. And lastly, the last terms of both the brackets are multiplied, so L = bd. In other words, every term of one binomial is multiplied with each term of the other binomial and then simplified by collecting the like terms together and adding them up.

For example 2: Multiply (6x + 5) and (3x + 2)
Solution:
Step 1: First = F = 6x * 3x = 18x^2
Step 2: Outer = O = 6x * 2 = 12x
Step 3: Inner = I = 5 * 3x = 15x
Step 4: Last = L = 5 * 2 = 10
Step 5: Collect and combine like terms: 18x^2 + (12x + 15x) + 10
Answer: 18x^2 + 27x +10

3.

Divide binomials:

Dividing binomials is rather simple. We write the division problem as a rational fraction and then cross cancel the common factors. If there are no common terms to cancel, then the rational fraction itself is? our answer.

For example 3: Divide 12a^2 + 6a by 6a^2 + 3
Solution:
Step 1: (12a^2 + 6a)/(6a^2+3)
Step 2:? = (6a(2a+1))/(3(2a^2+1)) now canceling common factors we have
Answer : 2a(2a+1)/(2a^2+1)

The Binomial Theorem:

The binomial theorem can be stated as follows:
(a+b)^n = nC0 * a^n + nC1 * a^(n-1)b + nC2 * a^(n-2)b^2 + ?? + nCr * a^(n-r)b^r + ??. + nCn * b^n

Source: http://article-marketing.ezinemark.com/operations-on-binomials-7d37569c42f9.html

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