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 :

9x^{2}, 5xy, - 4xy^{2}, 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 :

3x^{3} + x^{2} - 2 and 2x^{2} + 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 2x^{2} of the second polynomial below the corresponding term x^{2} of the first polynomial.

(Because x^{2} and 2x^{2} 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 x^{3} 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 + 7d^{2}) from (5c - d^{2}).

**Solution : **

Alternatively, this can also be done as :

(5c - d^{2}) - (3c + 7d^{2}) = 5c - d^{2} - 3c - 7d^{2 }

= (5c - 3c) + (-d^{2} - 7d^{2})

= 2c + (-8d^{2})

= 2c - 8d^{2}

**Example 5 :**

Subtract (2x^{2} + 2y^{2} - 6) from (3x^{2} - 7y^{2} + 9)

**Solution : **

**Example 6 :**

Add :

(7p^{3} + 4p^{2}- 8p + 1) and (3p^{3}- 5p^{2}- 10p + 5)

**Solution : **

= (7p^{3} + 4p^{2}- 8p + 1) + (3p^{3}- 5p^{2}- 10p + 5)

= 7p^{3} + 4p^{2}- 8p + 1 + 3p^{3}- 5p^{2}- 10p + 5

Group like terms together.

= (7p^{3} + 3p^{3}) + (4p^{2}- 5p^{2}) + (-8p - 10p) + (1 + 5)

Combine like terms.

= 10p^{3} - p^{2} -18p + 6

**Example 7 :**

Add :

2(x^{3} - x^{2} + 6x - 2) and (5x^{6} + 7x^{5} - 3x - 3)

**Solution : **

= 2(x^{3} - x^{2} + 6x - 2) + (5x^{6} + 7x^{5} - 3x - 3)

Distributive property.

= 2x^{3} - 2x^{2} + 12x - 4 + 5x^{6} + 7x^{5} - 3x - 3

Group like terms together.

= 5x^{6} + 7x^{5} + 2x^{3} - 2x^{2} + (12x - 3x) + (-4 - 3)

Combine like terms.

= 5x^{6} + 7x^{5} + 2x^{3} - 2x^{2} + 9x - 7

**Example 8 :**

Add :

-1(x^{6} + x^{3} + 6x^{2} - 2) and 2(5x^{6} + 7x^{5} - 3x - 3)

**Solution : **

= -1(x^{6} + x^{3} + 6x^{2} - 2) + 2(5x^{6} + 7x^{5} - 3x - 3)

Distributive property.

= -x^{6} - x^{3} - 6x^{2} + 2 + 10x^{6} + 14x^{5} - 6x - 6

Group like terms together.

= (-x^{6} + 10x^{6}) + 14x^{5} - x^{3} - 6x^{2} - 6x + (2 - 6)

Combine like terms.

= 9x^{6} + 14x^{5} - x^{3} - 6x^{2} - 6x - 4

**Example 9 :**

Subtract (12x^{3} + 14x^{2} + 17x - 12) from (15x^{3 }+ 22x^{2} + 17x - 19).

**Solution :**

= (15x^{3} + 22x^{2} + 17x - 19) - (12x^{3} + 14x^{2} + 17x - 12)

Distributive Property.

= 15x^{3} + 22x^{2} + 17x - 19 - 12x^{3} - 14x^{2} - 17x + 12

Group like terms together.

= (15x^{3} - 12x^{3}) + (22x^{2} - 14x^{2}) + (17x - 17x) + (-19 + 12)

Combine like terms.

= 3x^{3} + 8x^{2} + 0x - 7

= 3x^{3} + 8x^{2} - 7

**Example 10 :**

Subtract (5x^{3} + 3x^{2} + 7x - 6) from (3x^{3} + 2x^{2 }+ 6x - 4).

**Solution :**

= (3x^{3} + 2x^{2} + 6x - 4) - (5x^{3} + 3x^{2} + 7x - 6)

Distributive Property.

= 3x^{3} + 2x^{2} + 6x - 4 - 5x^{3} - 3x^{2} - 7x + 6

Group like terms together.

= (3x^{3} - 5x^{3}) + (2x^{2} - 3x^{2}) + (6x - 7x) + (-4 + 6)

Combine like terms.

= -2x^{3} - x^{2} - x + 2

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