Adding and Subtracting Polynomials

Our polynomial calisthenics begin today with adding and subtracting. Like any exercise, we need to do it correctly for it to help. Thankfully, our polynomial friends promise to share their little tricks that make things much easier.

The first trick is the ability to recognize and collect like terms. Like terms are little pieces of polynomials that have the same variables with the same exponents, but may differ in their numerical coefficients out front. For example, 2x and 3x are like terms, and 4ab² and 2ab² are like terms, too. Like terms are similar to paintings: they may look different up front, but they all look the same in the back.

To add or subtract like terms, we simply add or subtract the numerical coefficients. The trick here is we can only change the numerical coefficients out in front. Don't get carried away and start playing with the exponents.

The second little trick is our ability to remove parentheses. Remember our discoveries from our earlier visit to the planet of polynomials? If a set of parentheses has a "+" sign in front of it, the parentheses can simply be removed. Like this:

2x + (3x – 4y) =

2x + 3x – 4y

If the parentheses have a "–" sign in front, we need to distribute a -1 to every term inside the parentheses.

2x – (3x – 4y)

Okay, mathronaut. Remove the parentheses and distribute our -1.

2x – 3x + 4y

Time for a good workout with our little friends to build some muscles.

Sample Problem

Add 2x – 6y – z and 3x + 7y – 2z.

First, we rewrite it purely like a math problem. There are no words allowed between our numbers. We're being strict with our workout.

2x – 6y – z + 3x + 7y – 2z

Now we look for like terms and group 'em together.

(2x + 3x) + (-6y + 7y) + (-z – 2z) =

5x + y – 3z

Excellent. We can feel the burn already.

Sample Problem

Add 5m²ⁿ + 4mn – 8mn² and 2m²ⁿ – mn – mn².

Careful. We're excited, but we can't touch the exponents. They can be touchy at times. We can only add or subtract the numerical coefficients.

(5m²ⁿ + 2m²ⁿ) + (4mn – mn) + (-8mn² – mn²) =

7m²ⁿ + 3mn – 9mn²

Great. We're ready to ramp up the intensity to subtracting.

Sample Problem

Subtract 4a² – 7ab + b² from 2a² – ab – 6b².

Put it all together.

2a² – ab – 6b² – (4a² – 7ab + b²)

Remember that we're subtracting the entire first expression from the second, so we need to include those parentheses. Now, mathronauts, we remove the parentheses.

Easy, just like we learned in polynomial boot camp. Distribute that negative sign.

2a² – ab – 6b² – 4a² + 7ab – b²

Now to collect like terms, and remember: no touching the exponents allowed. We're serious about that. The last time someone poked an exponent during our workout, it pushed over the exercise bike and started a fight with the polynomials. Never again.

(2a² – 4a²) + (-ab + 7ab) + (-6b² – b²) =

-2a² + 6ab – 7b²

Good job, team. Now we and our little polynomials friends are ready for a challenge: simplifying.

Sample Problem

Simplify 3x – [2y – (7z – x – y) – 4x + 3z].

We can do this. We just need to remember what we've learned about parentheses, and lifting.

Always start with the inner parentheses, and then we'll work our way to the outside ones.

3x – [2y – 7z + x + y – 4x + 3z] 

So far, so good. Now we need to remove the outer braces, switching everything's sign as we go because of the "–" sign out front.

3x – 2y + 7z – x – y + 4x – 3z

Not too shabby. We're almost done. We just need to collect the like terms.

(3x – x + 4x) + (-2y – y) + (7z – 3z) =

6x – 3y + 4z

Excellent. Man, look at us. We're totally ripped from all that polynomial exercise. It's now time to move on to the next set—multiplying.