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 + 5y^{2}) from (5x - y^{2}).

**Solution : **

**= (5x - y**^{2}**) - (4x + 5y**^{2}**)**

**Distributive Property. **

**= ****5x - y**^{2}** - 4x - 5y**^{2}

**Group like terms together. **

**= (****5x - 4x) + (-y**^{2}** - 5y**^{2})

Combine like terms.

**= x - 6y ^{2}**

**Example 3 :**

Subtract (2x^{2} + 2y^{2} - 6) from (3x^{2} - 7y^{2} + 9).

**Solution : **

= (3x^{2} - 7y^{2} + 9) - (2x^{2} + 2y^{2} - 6)

Distributive Property.

= 3x^{2} - 7y^{2} + 9 - 2x^{2} - 2y^{2} + 6

Group like terms together.

= (3x^{2} - 2x^{2}) + (-7y^{2 }- 2y^{2}) + (9 + 6)

Combine like terms.

= x^{2} - 9y^{2} + 15

**Example 4 :**

Subtract (x^{3} + 4x^{2} - x + 6) from (2x^{3} + 5x^{2} - 2x + 7).

**Solution : **

= (2x^{3} + 5x^{2} - 2x + 7) - (x^{3} + 4x^{2} - x + 6)

**Distributive Property. **

= 2x^{3} + 5x^{2} - 2x + 7 - x^{3} - 4x^{2} + x - 6

Group like terms together.

= (2x^{3} - x^{3}) + (5x^{2} - 4x^{2}) + (-2x + x) + (7 - 6)

= x^{3} + x^{2} - x + 1

**Example 5 :**

Subtract (3 + 3x - 7x^{5} - 5x^{6}) from 2(x^{3} - x^{2} + 6x - 2).

**Solution : **

= 2(x^{3} - x^{2} + 6x - 2) - (3 + 3x - 7x^{5} - 5x^{6})

**Distributive Property. **

= 2x^{3} - 2x^{2} + 12x - 4 - 3 - 3x + 7x^{5} + 5x^{6}

Group like terms together.

= 5x^{6} + 7x^{5} + 2x^{3} - 2x^{2} + (12x - 3x) + (-4 - 3)

Combine like terms.

= 5x^{6} + 7x^{5} + 2x^{3} - 2x^{2} + 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 :

-0.03x^{2} + 26x - 1600

Western Plant :

-0.02x^{2} + 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

= (-0.03x^{2} + 26x - 1600) - (-0.02x^{2} + 20x - 1800)

Distributive Property.

= -0.03x^{2} + 26x - 1600 + 0.02x^{2} - 20x + 1800

Group like terms together.

= (-0.03x^{2} + 0.02x^{2}) + (26x - 20x) + (-1600 + 1800)

Combine like terms.

= -0.01x^{2} + 6x + 200

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