COMBINING LIKE TERMS
About "Combining like terms"
Combining like terms :
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.
Examples : 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.
Examples : 9x², 5xy, - 4xy², y, 6
More clearly,
Why should we know the difference between like terms and unlike terms ?
After having understood like terms, unlike terms and their difference, students have the question, "Why should we know the difference between like and unlike terms?" or "What is the use of knowing the difference between like terms and unlike terms?".
To do simplification (addition and subtraction) of algebraic expression, we have to combine like terms. So we have to know the difference between like and unlike terms.
Because, in algebraic expression, we can do addition and subtraction only on like terms not on unlike terms.
We
hope, now the students would have understood the reason for why we
should know the difference between the like and unlike terms.
Combing like terms
Adding and subtracting polynomials are nothing but combining the like terms. Here we give step by step explanation to combine like terms.
Let us look at some examples to have better understanding on combining like terms.
Example 1 :
Add : 3x³ + x² - 2 and 2x² + 5x + 5
Solution :
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 annexe the terms with zero coefficients.
Let us look at the next example on "Combining like terms"
Example 2 :
Find out the sum of the polynomials 3x - y, 2y - 2x, and x + y.
Solution :
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.
Let us look at the next example on "Combining like terms"
Example 3 :
Subtract 5xy from 8xy.
Solution :
Let us look at the next example on "Combining like terms"
Example 4 :
Subtract (3c + 7d²) from (5c - d²)
Solution :
Alternatively, this can also be done as :
( 5c - d² ) - ( 3c + 7d² ) = 5c - d² - 3c - 7d²
= ( 5c - 3c ) + ( -d² - 7d² )
= 2c + ( - 8d² )
= 2c - 8d²
Let us look at the next example on "Combining like terms"
Example 5 :
Subtract ( 2x² + 2y² - 6 ) from ( 3x² - 7y² + 9 )
Solution :
Let us look at the next example on "Combining like terms"
Example 6 :
Add ( 7p³ + 4p²- 8p + 1 ) and (3p³- 5p²- 10p + 5)
Solution :
Step 1:
The two given polynomials are already in the arranged form.So we can leave it as it is.
= ( 7p³ + 4p²- 8p + 1) + (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
Let us look at the next example on "Combining like terms"
Example 7 :
Add ( 2x³ + 5x² - 2x + 7 ) and ( x³ + 4x² - x + 6)
Solution :
= ( 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
Let us look at the next example on "Combining like terms"
Example 8 :
Add ( 3x³ - 2x² - x + 4 ) and ( 2x³ + 7x² - 3x - 3 )
Solution :
= (3x³ - 2x² - x + 4) + (2x³ + 7x² - 3x - 3)
= 3 x³ - 2 x² - x + 4 + 2 x³ + 7 x² - 3 x - 3
= 3x³ + 2x³ - 2x² + 7x² - x - 3x + 4 - 3
= 5x³ + 5x² - 4x + 1
Let us look at the next example on "Combining like terms"
Example 9 :
Add 2( x³ - x² + 6x - 2 ) and ( 5x⁶ + 7x⁵ - 3x - 3 )
Solution :
= 2( x³ - x² + 6 x - 2 ) + ( 5 x⁶ + 7 x⁵ - 3 x - 3 )
= 2x³ - 2x² + 12x - 4 + 5x⁶ + 7x⁵ - 3x - 3
= 5x⁶ + 7x⁵ + 2x³ - 2x² + 12x - 3x - 4 - 3
= 5x⁶ + 7x⁵ + 2x³ - 2x² + 9x - 7
Let us look at the next example on "Combining like terms"
Example 10 :
Add -1( x⁶ + x³ + 6x² - 2 ) and 2( 5x⁶ + 7x⁵ - 3x - 3 )
Solution :
= -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
Let us look at the next example on "Combining like terms"
Example 11 :
Add 5( 5x⁶ + 2x³ - 6x² - 2 ) + 6(-3x⁶ + 2x⁵ + 2x + 1 )
Solution :
= 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
Let us look at the next example on "Combining like terms"
Example 12 :
Add -2 ( 2x⁴ - 2x³ - x² + 5 ) and 3 ( 2x⁴ - 2x² - 3 )
Solution :
= -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
Let us look at the next example on "Combining like terms"
Example 13 :
Add 5( x⁴ - x³ + 5 ) and 2( x⁴ - 5x² - 7 )
Solution :
= 5( x⁴ - x³ + 5 ) + 2( x⁴ - 5x² - 7 )
= 5x⁴ - 5 x³ + 25 + 2x⁴ - 10x² - 14
= 5x⁴ + 2x⁴ - 5x³ - 10x² + 25 - 14
= 7x⁴ - 15x³ + 11
Let us look at the next example on "Combining like terms"
Example 14 :
Add 3( 6x⁴ - 2x³ - 3 ) and 2( 2x⁴ - x² - 8 )
Solution :
= 3 ( 6x⁴ - 2x³ - 3 ) + 2 ( 2x⁴ - x² - 8 )
= 18x⁴ - 6x³ - 9 + 4x⁴ -2x² - 16
= 18x⁴ + 4x⁴ - 6x³ -2x² - 9 - 16
= 22 x⁴ - 6x³ -2x² - 25
Let us look at the next example on "Combining like terms"
Example 15 :
Add (6x⁷-2x⁶-3x³+2x²) and 2(2x⁴+5x⁷+ 3x⁶+ x³+x²)
Solution :
= ( 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
Let us look at the next example on "Combining like terms"
Example 16 :
Add (x⁷-3x⁶-2x³+x²) and 5 (3x⁴+15x⁷+4x⁶+2x³+6x²)
Solution :
= (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
After having gone through the stuff and example problems, we hope that the students would have understood "How to combine like terms".
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Like fractions and unlike fractions
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