Adding and Subtracting Polynomials

Date: 12/08/96 at 13:57:10
From: Anonymous
Subject: Adding and subtracting polynomials

Dear Dr. Math,

Could you pleeeease explain to me how to add and subtract polynomials?  
I'm having so much trouble with them!

Thanks,

James Todd

Date: 12/10/96 at 16:18:07
From: Doctor Lisa
Subject: Re: Adding and subtracting polynomials

Hi James!

I'm just getting ready to teach this to my students tomorrow, so 
you're in luck!  I happen to have some examples right here.

Add the following:

(4x - 2x^2 - 7xy) + (2x^2 + 5xy)

When you are adding (or subtracting polynomials), you must find the 
variables and exponents that match.  For example, in the problem 
above, -2x^2 and 2x^2 have the same variable and exponent.  So do 
-7xy and 5xy.  Both the variables and the exponents each variable has 
must match exactly or you can't add (or subtract) them.  It's just 
like if you tried to add x and y -- you couldn't do it because they 
are different.  Once you find that something matches (like -2x^2 and 
2x^2), then you add the coefficients (remember, that's the number in 
front).  So this is what happens:

4x: there is nothing to add with it, so we leave it alone, but include 
it in the answer -2x^2 + 2x^2  = 0x^2, so we won't have an x^2 term in 
the answer -7xy + 5xy = -2xy

So the answer is 4x - 2xy.

Here's another with addition:

(4 - 5x^2 + 7x^3) + (4x^3 + 5x^2 + 5x^4)

We usually start with the highest power of the variable and work our 
way down.  Here goes:

5x^4: nothing to add to it, so we leave it be
7x^3 + 4x^3 = 11x^3
-5x^2 + 5x^2 = 0x^2 (so there will be no x^2 term in our answer)
4: nothing to add to it, so we leave it alone

The answer is 5x^4 + 11x^3 + 4.

Subtraction is very similar to addition.  You can think of it in one 
of two ways.  You can think of it as subtracting the coefficients 
(instead of adding them) OR you can think of it as adding the negative 
of the coefficients.

Here are a couple of subtraction problems:

(8x^3 + x^2 - 7x - 11) - (5x^3 + 3x^2 - 3x + 8)

When I teach this to my students, I tell them to go through and 
distribute the negative sign to the second group.  I tell them to do 
this because sometimes when you go to subtract, there isn't a like 
term in the first group.  However, you still need to make the sign in 
front of the number you're trying to subtract the opposite of what you 
are given.  By distributing the negative, you make sure to properly 
take care of these terms.

So I would do this first:

(8x^3 + x^2 - 7x - 11) - 5x^3 - 3x^2 + 3x - 8

Now we have a situation like the one we had before with addition:

8x^3 - 5x^3 = 3x^3
x^2 - 3x^2 = -2x^2
-7x + 3x = -4x
-11 - 8 = -19

So the answer is: 3x^3 - 2x^2 - 4x - 19

Here's another:

(4a^2 - 6a) - (2a^2 + 5a - 3)

Taking care of the negative I get:

(4a^2 - 6a) - 2a^2 - 5a + 3

Combining like terms:

4a^2 - 2a^2 = 2a^2
-6a - 5a = -11a
3: this stays as 3 since there is nothing to add to in the first set

So the answer is:  2a^2 - 11a + 3

I hope this helps you out!

-Doctor Lisa,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/