Adding and Subtracting Polynomials
Adding and Subtracting Polynomials
Adding polynomials is just a matter of combining like terms, with some order of operations
considerations thrown in. As long as you're careful with the minus signs, and don't confuse addition and
multiplication, you should do fine. There are a couple formats for adding and subtracting, and they
hearken back to earlier times, when you were adding and subtracting just plain old numbers. First, you
learned addition "horizontally", like this: 6 + 3 = 9. You can add polynomials in the same way,
grouping like terms and then simplifying. Simplify (2x + 5y) + (3x - 2y) I'll clear the parentheses,
group like terms, and then simplify:
(2x + 5y) + (3x - 2y)
= 2x + 5y + 3x - 2y
= 2x + 3x + 5y - 2y
= 5x + 3y
Horizontal addition works fine for simple examples. But when you were adding plain old numbers, you
didn't generally try to add 432 and 246 horizontally; instead, you would "stack" them vertically, one on
top of the other, and then add down the columns: was the x3 column, the second column was the x2
column, the third column was the x column, and the fourth column was the constants column.
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You can do the same thing with polynomials. This is how the above simplification exercise looks when
it is done "vertically": Simplify (2x + 5y) + (3x - 2y) , I'll put each variable in its own column; in this
case, the first column will be the x-column, and the second column will be the y-column: I get the same
solution vertically as I got horizontally: 5x + 3y. The format you use, horizontal or vertical, is a matter
of taste (unless the instructions explicitly tell you otherwise). Given a choice, you should use whichever
format that you're more comfortable and successful with. Note that, for simple additions, horizontal
addition (so you don't have to rewrite the problem) is probably simplest, but, once the polynomials get
complicated, vertical is probably safest bet (so you don't "drop", or lose, terms and minus signs).
Subtracting polynomials is quite similar to adding polynomials, but you have that pesky minus sign to
deal with. Here are some examples, done both horizontally and vertically:
Simplify (x3 + 3x2 + 5x - 4) - (3x3 - 8x2 - 5x + 6)
The first thing I have to do is take that negative through the parentheses. Some students find it helpful
to put a "1" in front of the parentheses, to help them keep track of the minus sign:
Horizontally:
(x3 + 3x2 + 5x - 4) - (3x3 - 8x2 - 5x + 6)
= (x3 + 3x2 + 5x - 4) - 1(3x3 - 8x2 - 5x + 6)
= (x3 + 3x2 + 5x - 4) - 1(3x3) - 1 (-8x2) - 1(-5x) - 1(6)
= x3 + 3x2 + 5x - 4 - 3x3 + 8x2 + 5x - 6
= x3 - 3x3 + 3x2 + 8x2 + 5x + 5x - 4 - 6
= -2x3 + 11x2 + 10x -10
Vertically: Copyright (c) Elizabeth Stapel 2000-2011 All Rights Reserved
In the horizontal case, you may have noticed that running the negative through the parentheses changed
the sign on each term inside the parentheses. The shortcut here is to not bother writing in the subtaction
sign or the parentheses; instead, you just change all the signs in the second row.
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