ADDING AND SUBTRACTING POLYNOMIALS

About "Adding and subtracting polynomials"

On the webpage adding and subtracting polynomials examples with answers, we are going to see how to add and subtract two 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

Example :

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

Subtracting polynomials

Example  :

Subtract the following polynomials:

(2 x³ - 2 x² + 4 x - 3)- (x³ + x² - 5 x + 4)

Solution :

Step 1:

In the first step we are going to multiply the negative with inner terms.

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

Step 2:

In the second step we have to combine the like terms 

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

Step 3:

After combining the like terms we will get the answer

                  = x³ - 3 x² + 9x - 7

Adding and subtracting polynomials - Examples

Example 1 :

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 next example on "adding and subtracting polynomials"

Example 2 :

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 next example on "Adding and subtracting polynomials"

Example 3 :

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 next example on "Adding and subtracting polynomials"

Example 4 :

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 next example on "Adding and subtracting polynomials"

Example 5 :

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 next example on "Adding and subtracting polynomials"

Example 6 :

Subtract 2x³ + 5x² - 2x - 11 from 3x³ - 2x² - 5x - 6

Solution :

= ( 3x³ - 2x² - 5x - 6 ) - ( 2x³ + 5x² - 2x - 11 )

   = 3x³ - 2x² - 5x - 6 -2x³ - 5x² + 2x + 11

   = 3x³ - 2x³ - 2x² - 5x² - 5x + 2x - 6 + 11

   = x³ - 7x² - 3x + 5

Question 7 :

Subtract x³ + 4x² - 12x - 5  from 5x³ + 3x² + 2x - 10

Solution : 

 = ( 5x³ + 3x² + 2x - 10 ) - ( x³ + 4x² - 12x - 5 )

   = 5x³ + 3x² + 2x - 10 - x³ - 4x² + 12x + 5 

   = 5x³ - x³ + 3x² - 4x² + 2x + 12x - 10 + 5 

   = 4x³ - x² + 14x - 5

Example 8 :

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

Solution :

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

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

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

   = 3x³ + 8x² + 0x - 7

   = 3x³ + 8x² - 7

Example 9 :

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

Solution :

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

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

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

   = -2x³ - x² - x + 2

Example 10 :

Subtract x³ + 32x² + 17x - 16 from 13x³+23x²+16x-14

Solution :

 = ( 13x³ + 23x² + 16x - 14 ) - ( x³ + 32x² + 17x - 16 )

   = 13x³ + 23x² + 16x - 14 - x³ - 32x² - 17x + 16

   = 13x³ - x³ + 23x² - 32x² + 16x - 17x - 14 + 16

   = 12x³ - 9x² - x + 2

After having gone through the stuff given above, we hope that the students would have understood "Adding and subtracting polynomials". 

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