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### Adding and Subtracting Polynomials

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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
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```
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.

(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

-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/
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Associated Topics:
High School Basic Algebra
High School Polynomials
Middle School Algebra

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