Adding Polynomials
Purplemath
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 polynomials, and they hearken back to the two methods you learned for addition and subtract of plain numbers, back when you were in grade school. First, you learned addition "horizontally", like this:
6 + 3 = 9
That is, you were given relatively small values, and you learned to do the addition — largely in your head, and by working horizontally. We can add polynomials in the same way, grouping any "like" terms and then simplifying the results.
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Simplify (2x + 5y) + (3x − 2y)
I'll clear the parentheses first. This is easy to do when adding, because there are no "minus" signs to take through any parentheticals. Then I'll group the like terms in accordance to their variables (keeping them in alphabetical order), and finally I'll simplify:
(2x + 5y) + (3x − 2y)
2x + 5y + 3x − 2y
2x + 3x + 5y − 2y
5x + 3y
These two terms are un-like (because they have different variables), so I cannot combine them. This means that I've gone as far as I can, so my hand-in answer is:
5x + 3y
Horizontal addition works fine for simple polynomials. But when you were adding plain old numbers, you didn't generally try to apply horizontal addition to adding numbers like 432 and 246; instead, you would stack the numbers vertically, one on top of the other, and then add down the columns (doing "carries", as necessary):
You can do the same thing with polynomials. Here's how the above simplification exercise looks, when it is done "vertically"
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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).
One advantage of vertical polynomial addition over vertical numerical addition: there is never anything to "carry" from one column to the next.
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Simplify (3x3 + 3x2 − 4x + 5) + (x3 − 2x2 + x − 4)
I can add horizontally:
(3x3 + 3x2 − 4x + 5) + (x3 − 2x2 + x − 4)
3x3 + 3x2 − 4x + 5 + x3 − 2x2 + x − 4
3x3 + x3 + 3x2 − 2x2 − 4x + x + 5 − 4
4x3 + 1x2 − 3x + 1
...or vertically:
Either way, I get the same answer. For my final hand-in answer, I'll remove the "understood" 1s.
4x3 + x2 − 3x + 1
Note that each column in the vertical addition above contains only one degree of x: the first column above (that is, the left-most column being added down) 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. This is analogous to having a thousands column, a hundreds column, a tens column, and a ones column when doing strictly-numerical addition.
And, just as we need to use zeroes to fill empty slots in hundreds columns (or whichever column has no digit), we need to leave spaces in vertical addition for any gaps in the powers of variables.
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Simplify (7x2 − x − 4) + (x2 − 2x − 3) + (−2x2 + 3x + 5)
It's perfectly okay to have to add three or more polynomials at once. I'll just go slowly and do each step throroughly, and it should work out right.
First, I'll do the adding horizontally:
(7x2 − x − 4) + (x2 − 2x − 3) + (−2x2 + 3x + 5)
7x2 − x − 4 + x2 − 2x − 3 + −2x2 + 3x + 5
7x2 + 1x2 − 2x2 − 1x − 2x + 3x − 4 − 3 + 5
8x2 − 2x2 − 3x + 3x − 7 + 5
6x2 − 2
Note the 1's in the third line. Any time I have a variable without a coefficient, there is an "understood" 1 as the coefficient. If it's helpful to me to write that 1 in, then I'll do so.
Now, I'll do the adding vertically:
Either way, I get the same answer. For my hand-in answer, I won't include the "+0x" term.
6x2 − 2
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Simplify (x3 + 5x2 − 2x) + (x3 + 3x − 6) + (−2x2 + x − 2)
Horizontally:
(x3 + 5x2 − 2x) + (x3 + 3x − 6) + (−2x2 + x − 2)
x3 + 5x2 − 2x + x3 + 3x − 6 + −2x2 + x − 2
x3 + x3 + 5x2 − 2x2 − 2x + 3x + x − 6 − 2
2x3 + 3x2 + 2x − 8
When I add large numbers, there are sometimes zeroes in the numbers, such as in the following:
The zeroes in "1002" stand for "zero hundreds" and "zero tens". They are what is called "placeholders", indicating that there are no hundreds or tens. If I didn't include those zeroes in the numerical expression, then I'd have (in the top line) "12", which isn't what I mean. The zeroes keep things lined up properly. When I vertically add polynomials that skip some of the degrees of x, I need to leave gaps, so that the terms in the various polynomials line up properly (that is, according to degree).
Here's what it looks like, when I have polynomials with gaps in their powers, and I add vertically:
Whether working vertically or horizontally, I get the same answer:
2x3 + 3x2 + 2x − 8
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