In order to simplify (addition or subtraction) algebraic expressions, we need to combine like terms.
Before we do combining like terms, first let us come to know what are like terms and unlike terms.
Like Terms or Similar Terms:
Like terms are the terms which have the same variables with same exponent for each variable.
7x, 3x, - 4x
Unlike Terms or Dissimilar Terms:
Unlike terms are the terms which have same variables or different variables.
If they have same variables, the exponents will not be same.
9x2, 5xy, - 4xy2, y, 6
To do simplification (addition and subtraction) of algebraic expression, we can combine only like terms, not unlike terms.
Example 1 :
3x3 + x2 - 2 and 2x2 + 5x + 5
First we have to arrange the like terms column wise and add them.
Observe the important points related to the above work.
1. We have written the term 2x2 of the second polynomial below the corresponding term x2 of the first polynomial.
(Because x2 and 2x2 are like terms and they can be combined)
2. Similarly, the constant term +5 is placed below the constant term – 2.
3. Since the term x in the first polynomial and the term x3 in the second polynomial do not exist, their respective places have been left blank to facilitate the process of addition. Or, for the non existing terms, we annex the terms with zero coefficients.
Example 2 :
Find out the sum of the polynomials :
3x - y, 2y - 2x, and x + y.
We can find out the sum of the given polynomials using either column method of addition or row method of addition as explained below.
Once they are written in the above methods, we have to combine the like terms.
Example 3 :
Subtract 5xy from 8xy.
Example 4 :
Subtract (3c + 7d2) from (5c - d2).
Alternatively, this can also be done as :
(5c - d2) - (3c + 7d2) = 5c - d2 - 3c - 7d2
= (5c - 3c) + (-d2 - 7d2)
= 2c + (-8d2)
= 2c - 8d2
Example 5 :
Subtract (2x2 + 2y2 - 6) from (3x2 - 7y2 + 9)
Example 6 :
(7p3 + 4p2- 8p + 1 ) and (3p3- 5p2- 10p + 5)
Step 1 :
The two given polynomials are already in the arranged form.So we can leave it as it is.
= (7p3 + 4p2- 8p + 1 ) and (3p3- 5p2- 10p + 5)
Step 2 :
Now we have to write the like terms together starting from the highest power to lowest power.
= 7p3 + 3p3 + 4p2- 5p2- 8p - 10p + 1 + 5
So, the final answer is
10p3- 1p2- 18p + 6.
Example 7 :
(2x3 + 5x2 - 2x + 7) and (x3 + 4x2 - x + 6)
= (2x3 + 5x2 - 2x + 7) + (x3 + 4x2 - x + 6)
= 2x3 + 5x2 - 2x + 7 + x3 + 4x2 - x + 6
= 2x3 + x3 + 5x2 + 4x2 - 2x - x + 7 + 6
= 3x3 + 9x2 - 3x + 13
Example 8 :
(3x3 - 2x2 - x + 4) and (2x3 + 7x2 - 3x - 3)
= (3x3 - 2x2 - x + 4) + (2x3 + 7x2 - 3x - 3)
= 3x3 - 2x2 - x + 4 + 2x3 + 7x2 - 3 x - 3
= 3x3 + 2x3 - 2x2 + 7x2 - x - 3x + 4 - 3
= 5x3 + 5x2 - 4x + 1
Example 9 :
2(x3 - x2 + 6x - 2) and (5x6 + 7x5 - 3x - 3)
= 2(x3 - x2 + 6x - 2) + (5x6 + 7x5 - 3x - 3)
= 2x3 - 2x2 + 12x - 4 + 5x6 + 7x5 - 3x - 3
= 5x6 + 7x5 + 2x3 - 2x2 + 12x - 3x - 4 - 3
= 5x6 + 7x6 + 2x3 - 2x2 + 9x - 7
Example 10 :
-1(x6 + x3 + 6x2 - 2) and 2(5x6 + 7x5 - 3x - 3)
= -1(x6 + x3 + 6x2 - 2) + 2(5x6 + 7x5 - 3x - 3)
= -x6 - x3 - 6x2 + 2 + 10x6 + 14x5 - 6x - 6
= -x6 + 10x6 + 14x5 - x3 - 6x2 - 6x + 2 - 6
= 9x6 + 14x5 - x3 - 6x2 - 6x - 4
Example 11 :
5(5x6 + 2x3 - 6x2 - 2) + 6(-3x6 + 2x5 + 2x + 1)
= 5(5x6 + 2x3 - 6x2 - 2) + 6(-3x6 + 2x5 + 2x + 1)
= 25x6 + 10x3 - 30x2 - 10 -18x6 + 12x5 + 12x + 6
= 25x6 -18x6 + 12x5 + 10x3 - 30x2 + 12x -10 + 6
= 7x6 + 12x5 + 10x3 - 30x2 + 12x - 4
Example 12 :
-2(2x4 - 2x3 - x2 + 5) and 3(2x4 - 2x2 - 3)
= -2(2x4 - 2x3 - x2 + 5) + 3(2x4 - 2x2 - 3)
= -4x4 + 4x3 + 2x2 -10 + 6x4 - 6x2 - 9
= -4x4 + 6x4 + 4x3 + 2x2 - 6x2 -10 - 9
= 2x4 + 4x3 - 4x2 - 19
Example 13 :
5(x4 - x3 + 5) and 2(x4 - 5x2 - 7)
= 5(x4 - x3 + 5) + 2(x4 - 5x2 - 7)
= 5x4 - 5x3 + 25 + 2x4 - 10x2 - 14
= 5x4 + 2x4 - 5x3 - 10x2 + 25 - 14
= 7x4 - 5x3 - 10x2 + 11
Example 14 :
3(6x4 - 2x3 - 3) and 2(2x4 - x2 - 8)
= 3(6x4 - 2x3 - 3) + 2(2x4 - x2 - 8)
= 18x4 - 6x3 - 9 + 4x4 - 2x2 - 16
= 18x4 + 4x4 - 6x3 - 2x2 - 9 - 16
= 22x4 - 6x3 -2x2 - 25
Example 15 :
(6x7 - 2x6 - 3x3 + 2x2) and 2(2x4 + 5x7 + 3x6 + x3 + x2)
= (6x7 - 2x6 - 3x3 + 2x2) + 2(2x4 + 5x7 + 3x6 + x3 + x2)
= 6x7 - 2x6 - 3x3 + 2x2 + 4x4 + 10x7 + 6x6 + 2x3 + 2x2
= 6x7 + 10x7- 2x6 + 6x6 + 4x4 - 3x3 + 2x3 + 2x2 + 2x2
= 16x7 + 4x6 + 4x4 - x3 + 4x2
Example 16 :
(x7 - 3x6 - 2x3 + x2) and 5(3x4 + 15x7 + 4x6 + 2x3 + 6x2)
= (x7 - 3x6 - 2x3 + x2) + 5(3x4 + 15x7 + 4x6 + 2x3 + 6x2)
= x7 - 3x6 - 2x3 + x2 + 15x4 + 75x7 + 20x6 + 10x3 + 30x2
= x7 + 75x7- 3x6 + 20x6 - 2x3 + 10x3 + x2 + 30x2
= 76x7 + 17x6 + 8x3 + 31x2
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