SUBTRACTING POLYNOMIALS

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) + (-7y² - 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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