Subtracting Polynomials

You may have noticed on the adding polynomials page that you had to subtract individual terms from one another. This is different from subtracting entire polynomials from one another, as you will soon see. Before you can learn how to subtract polynomials, you must first learn the Distributive Law.

The Distributive Law

When you have a polynomial in brackets, and you have anything OTHER than a single plus sign out front of the brackets, you must use the Distributive Law. Let's look at an example:

Distributive Law Example 1

Simplify the following expression: 3(2x + 3y + 7).

Solution

If you are following the order of operations, the first thing you should do is check inside the brackets to see if there is any simplifying to do. There is not (none of the terms are like terms so they cannot be combined), so you can turn your attention outside the brackets. Notice the 3. This means 3 times the polynomial found inside the brackets. In other words, you must:

multiply EACH TERM inside the brackets by whatever number is found outside the brackets.

That's why this is called the Distributive Law. You Distribute the number throughout the brackets evenly. To show this, see the picture below:

Once you have performed each operation, you can remove the brackets. Your final expression in this example would be:

6x + 9y + 21

At this point you might be asking "What does multiplication have to do with SUBTRACTING polynomials". Before we answer that question, look at the next Distributive Law example:

Distributive Law Example 2

Simplify the following expression: -(2x + 3y + 7).

Solution

Again, you should check inside the brackets to see if there is any simplification to be done (and there is none). Now look outside the front of the brackets. There is again something other than a plus sign. You see a negative sign (which is the same as a negative 1)! You must multiply the negative by each term inside the brackets just like any other number.

The effect of multiplying each term inside the brackets by a -1 is that the sign of each of terms will change to the opposite sign. Again, once you perform the multiplication you can eliminate the brackets. Your final expression will be:

-2x - 3y - 7

Now we'll try a sample polynomial subtraction question:

Polynomial Subtraction Example 1

Perform the following operation: (7a + 3b - 2) - (3a - 8b - 11).

Solution

First, look inside both sets of brackets to see if there is any simplification to be done. (There is none.) Now try to eliminate the brackets. You can get rid of the first set of brackets immediately as there is nothing out front to be distributed though the brackets. However, notice the negative outside the front of the second set of brackets. This must be distributed though the brackets. You end up with the expression:

7a + 3b -2 -3a + 8b + 11

Combine like terms to arrive at:

4a +11b +9

Polynomial Subtraction Example 2

Subtract (9x² + 11x +3) from (4x² + 6).

Solution

First, set this question up as an alebraic expression:

(4x² + 8) - (9x² + 11x + 3)

Distribute the negative though the second set of brackets and eliminate the brackets.

4x² + 8 - 9x² - 11x - 3

Now combine like terms to arrive at your final solution:

-5x² - 11x + 5

Polynomial Subtraction Example 3

Let's look at a longer example. Simplify the following expression:

a³ - 9 + 3(2a³ - 8a) - 2(a³ - 8a² + 3a - 9)

Solution

This is a good example of having to pay close attention to each step of your solution. First, check to see if there is any simplification needed INSIDE of the brackets. (Nope!) Now we need to get rid of those brackets. Both sets of brackets have a number out front that needs to be distributed through the brackets. After distributing, you arrive at this expression:

a³ - 9 + 6a³ - 24a - 2a³ + 16a² -6a + 18

To try and keep things straight, let's group all of the like terms together:

a³ + 6a³ - 2a³ + 16a² - 24a - 6a - 9 + 18

Now combine like terms to arrive at the final answer:

5a³ + 16a² - 30a + 9

Now that you have seen both adding polynomials and subtracting polynomials you can try some sample questions, or head back to the adding and subtracting polynomials main page.