Consider the two polynomials A and B.

You want to find (A - B).

That is, you want to subtract polynomial B from polynomial A. 

The following steps will be useful to do the above subtraction of polynomials. 

Step 1 :

To subtract the polynomial B from polynomial A, write the polynomials in the form :

A - B

Step 2 : 

Use Distribute Property. 

That is, distribute negative sign to all the terms of polynomial B. 

Step 3 :

Group like terms together. 

Step 3 :

Combine like terms. 

Example 1 :

Subtract 5ab from 8ab. 

Solution : 

=  8ab - 5ab

=  3ab

Example 2 :

Subtract (4x + 5y2)  from  (5x - y2).

Solution : 

=  (5x - y2) - (4x + 5y2)

Distributive Property. 

=  5x - y2 - 4x - 5y2

Group like terms together. 

=  (5x - 4x) + (-y2 - 5y2)

Combine like terms. 

=  x - 6y2

Example 3 :

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

Solution : 

=  (3x2 - 7y2 + 9) - (2x2 + 2y2 - 6)

Distributive Property. 

=  3x2 - 7y2 + 9 - 2x2 - 2y2 + 6

Group like terms together. 

=  (3x2 - 2x2) + (-7y- 2y2) + (9 + 6)

Combine like terms. 

=  x2 - 9y2 + 15

Example 4 :

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

Solution : 

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

Distributive Property. 

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

Group like terms together. 

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

=  x3 + x2 - x + 1

Example 5 :

Subtract (3 + 3x - 7x5 - 5x6) from 2(x3 - x2 + 6x - 2). 

Solution : 

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

Distributive Property. 

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

Group like terms together. 

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

Combine like terms. 

=  5x6 + 7x5 + 2x3 - 2x2 + 9x - 7

Example 6 :

The profits of two different production plants can be modeled as shown below, where x' is the number of units produced at each plant. 

Eastern Plant : 

-0.03x2 + 26x - 1600

Western Plant : 

-0.02x2 + 20x - 1800

Write a polynomial that represents the difference of the profits at the eastern plant and the profits at the western plant.  

Solution :

=  Eastern plant profits - Western plant profits

=  (-0.03x2 + 26x - 1600) - (-0.02x2 + 20x - 1800)

Distributive Property.

=  -0.03x2 + 26x - 1600 + 0.02x2 - 20x + 1800

Group like terms together. 

=  (-0.03x2 + 0.02x2) + (26x - 20x) + (-1600 + 1800)

Combine like terms. 

=  -0.01x2 + 6x + 200

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