ADDING POLYNOMIALS

To add two or more polynomials, we need to combine the 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.

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. 

Examples

Example 1 :

Add the following two polynomials. 

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 :

Add the following 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 :

Add the following two polynomials. 

(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 ) 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 4 :

Add the following two polynomials. 

(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

Example 5 :

Add the following two polynomials. 

(3x³ - 2x² - x + 4) and (2x³ + 7x² - 3x - 3)

Solution : 

=  (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 6 :

Add the following two polynomials. 

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

Solution : 

=  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 7 :

Add the following two polynomials. 

-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

Example 8 :

Add the following two polynomials. 

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

Example 9 :

Add the following two polynomials. 

-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

Example 10 :

Add the following two polynomials. 

5(x⁴ - x³ + 5) and  2(x⁴ - 5x² - 7)

Solution :

=  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 11 : 

Add the following two polynomials. 

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

=  22x⁴ - 6x³ -2x² - 25

Example 12 :

Add the following two polynomials. 

(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

Example 13 :

Add the following two polynomials. 

(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

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