COMBINING LIKE TERMS

To simplify (addition or subtraction) algebraic expressions, we need to combine like terms.

To combine like terms, we must be aware 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.

Example :

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.

Example :

9x², 5xy, - 4xy², y, 6

More clearly,

Note :

To do simplification (addition and subtraction) of algebraic expression, we can combine only like terms, not unlike 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 annex the terms with zero coefficients.

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.

Example 3 :

Subtract 5xy from 8xy.

Solution :

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

Example 5 :

Subtract (2x² + 2y² - 6)  from  (3x² - 7y² + 9).

Solution :

Example 6 :

Add :

(7p³ + 4p² - 8p + 1) and (3p³- 5p²- 10p + 5)

Solution :

= (7p³ + 4p²- 8p + 1) + (3p³- 5p²- 10p + 5)

= 7p³ + 4p²- 8p + 1 + 3p³- 5p²- 10p + 5

Group like terms together.

= (7p³ + 3p³) + (4p²- 5p²) + (-8p - 10p) + (1 + 5)

Combine like terms.

= 10p³ - p² -18p + 6

Example 7 :

Add :

2(x³ - x² + 6x - 2) and (5x⁶ + 7x⁵ - 3x - 3)

Solution :

= 2(x³ - x² + 6x - 2) + (5x⁶ + 7x⁵ - 3x - 3)

Distributive property.

= 2x³ - 2x² + 12x - 4 + 5x⁶ + 7x⁵ - 3x - 3

Group like terms together.

= 5x⁶ + 7x⁵ + 2x³ - 2x² + (12x - 3x) + (-4 - 3)

Combine like terms.

= 5x⁶ + 7x⁵ + 2x³ - 2x² + 9x - 7

Example 8 :

Add :

-1(x⁶ + x³ + 6x² - 2) and 2(5x⁶ + 7x⁵ - 3x - 3)

Solution :

= -1(x⁶ + x³ + 6x² - 2) + 2(5x⁶ + 7x⁵ - 3x - 3)

Distributive property.

= -x⁶ - x³ - 6x² + 2 + 10x⁶ + 14x⁵ - 6x - 6

Group like terms together.

= (-x⁶ + 10x⁶) + 14x⁵ - x³ - 6x² - 6x + (2 - 6)

Combine like terms.

= 9x⁶ + 14x⁵ - x³ - 6x² - 6x - 4

Example 9 :

Subtract (12x³ + 14x² + 17x - 12) from (15x³ + 22x² + 17x - 19).

Solution :

= (15x³ + 22x² + 17x - 19) - (12x³ + 14x² + 17x - 12)

Distributive Property.

= 15x³ + 22x² + 17x - 19 - 12x³ - 14x² - 17x + 12

Group like terms together.

= (15x³ - 12x³) + (22x² - 14x²) + (17x - 17x) + (-19 + 12)

Combine like terms.

= 3x³ + 8x² + 0x - 7

= 3x³ + 8x² - 7

Example 10 :

Subtract (5x³ + 3x² + 7x - 6) from (3x³ + 2x² + 6x - 4).

Solution :

= (3x³ + 2x² + 6x - 4) - (5x³ + 3x² + 7x - 6)

Distributive Property.

= 3x³ + 2x² + 6x - 4 - 5x³ - 3x² - 7x + 6

Group like terms together.

= (3x³ - 5x³) + (2x² - 3x²) + (6x - 7x) + (-4 + 6)

Combine like terms.

= -2x³ - x² - x + 2

Example 11 :

Simplify :

(a + b)(a - b) - (a² + b²)

Solution :

= (a + b)(a - b) - (a² + b²)

= a² - b² - (a² + b²)

= a² - b² - a²  - b²

= -2b²

Example 12 :

Simplify :

(a + b)³ - (a - b)³

Solution :

We can use the following algebraic identities to simplify the given expressions.

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a - b)³ = a³ - 3a²b + 3ab² - b³

(a + b)³ - (a - b)³ :

= (a³ + 3a²b + 3ab² + b³) - (a³ - 3a²b + 3ab² - b³)

= a³ + 3a²b + 3ab² + b³ - a³ + 3a²b - 3ab² + b³

= 3a²b + b³ + 3a²b + b³

= 6a²b + 2b³

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