Common Terms in Algebraic Expressions

In this video we shall learn simplification of algebraic expressions using common factors.  The following are the important terms associated with them.

Expressions: An algebraic expression is a phrase which contains numeric coefficients (constants), literal coefficients (variables) and arithmetic operators of either addition (+), subtraction (-), multiplication(x) and division(). For example, 12x2y + 15xy3 is an algebraic expression.

Terms: each individual part of an algebraic expression is called a term. For example, in the expression above there are two terms.  12x2y and 15xy3.

Factors: each algebraic term can be expressed in  the form of its lowest factors. For example,

12x2y  can be written as : 2 x 2 x 3 x x x x x y

15xy3 can be written as: 3 x 5 x x x y x y x y

To list the common factors in the terms above, we look at the common numeric and literal coefficients of both the terms.

The common coefficients are: 3, x and y

Thus, the common factor of 12x2y and 15xy3 is 3xy. We use the common factor to divide each of the terms. The answer so obtained, along with the common factor is the simplified form of the expression.

12x2y ÷ 3xy + 15xy3 ÷ 3xy = 4x+5y2

Thus, 12x2y + 15xy3 = 3xy (4x+5y2).

Let us look at a few examples:

  • 3y2+7y

Common factor: y

On dividing the common factor with the terms we get: 3y+7

3y2+7y  = y (3y+7)

  • 5x2y+15xy2

Common factor:  5xy

On dividing the common factor with the terms we get: x+3y

5x2y+15xy2 = 5xy (x+3y)

  • 2a2b-6bc+8abc

Common factor: 2b

On dividing the common factor with the terms we get: a2-3c+4ac

2a2b-6bc+8abc =2b (a2-3c+4ac)

  • Ax+ay+8x+8y

Common factor: (x+ y)

On dividing the common factor with the terms we get:  (x+y)(a+8)

Ax+ay+8x+8y = a(x+y) +8(x+y) = (x+y)(a+8)

  • xy-zx-2y+2z

Common factor:  (y-z)

On dividing the common factor with the terms we get: (y-z)(x-2)

xy-zx-2y+2z = x(y-z)-2(y-z) = (y-z)(x-2)

  • 3xy-5x+6x-10

Common factor:  (3y-5)

On dividing the common factor with the terms we get: (3y-5)(x+2)

3xy-5x+6x-10 = x(3y-5) +2(3y-5) = (3y-5)(x+2)

  • 12x2+ 1/24 (x5y2)

Common factor: x2  

On dividing the common factor with the terms we get: 12+(1/24)x3y2

12x2+ 1/24 (x5y2) = x2[12+(1/24)x3y2]

  • 4/9 (p2q) – 16pq4

Common factor:  4pq

On dividing the common factor with the terms we get: (1/q) p-4q3

4/9 (p2q) – 16pq4 = 4pq [(1/q) p-4q3]

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