ADDING POLYNOMIALS

How to add two polynomials?

Adding polynomials is nothing but combining the like terms.

Let us consider the following problem. 

Add : (3x³-5x²+ 2x-7) and (4x²+x-8)

Here we give step by step explanation for adding the above two polynomials.

Step 1 :

Before going to add two polynomials, first we have to arrange the given polynomials one by one from highest power to lowest power.

(3x³-5x²+ 2x-7) and (4x²+x-8)

The two given polynomials are already in the arranged form.So we can leave it as it is.

Step 2 :

Now we have to write the like terms together starting from the highest power to lowest power.

     = (3x³-5x²+ 2x-7) + (4x²+ x - 8)

Step 3:

Combine the like terms (Add or subtract) based on the signs of those terms.

In the second polynomial,we do not have x³ term,so we have to consider that there is zero x³. 

So the final answer is  3x³ - 1x² + 3x - 15

To have better understanding, let us look at some more examples on "Adding polynomials"

Example 1 :

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 "Adding polynomials"

Example 2 :

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 "Adding polynomials"

Example 3 :

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 "Adding polynomials"

Example 4 :

  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 "Adding polynomials"

Example 5 :

  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 "Adding polynomials"

Example 6 :

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 "Adding polynomials"

Example 7 :

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 "Adding polynomials"

Example 8 :

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 "Adding polynomials"

Example 9 : 

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 "Adding polynomials"

Example 10 :

 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 "Adding polynomials"

Example 11 :

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 examples explained above, we hope that the students would have understood the stuff "Adding polynomials"

Related Topics : 

Like terms and unlike terms

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