Polynomial long division functions similarly to long division, and if the
division leaves no remainder, then the divisor is called a factor.
LEARNING OBJECTIVE [ edit ]
Use polynomial division to find additional factors of a polynomial
KEY POINTS [ edit ]
Dividing one polynomial by another can be achieved by usinglong division. The rules for
polynomial long division are the same as the rules learned for long division of integers.
The four steps of long division are divide, multiply, subtract, and bring down.
After completing polynomial long division, it is good to check the answers, either by plugging in a
number or by multiplying the quotient times the divisor to get the dividend back.
TERMS [ edit ]
divisor
An integer that divides another integer an integral number of times.
quotient
The number resulting from the division of one number orexpression by another.
dividend
A number or expression that is to be divided by another.
Give us feedback on this content: FULL TEXT [edit ]
Simplifying, multiplying, dividing, adding, and subtractingrational expressionsare all based
on the basic skills of working with fractions. Dividing polynomials is based on an even earlier
skill, one that pretty much everyone remembers with horror: long division.
To refresh one's memory, try dividing 745
3
by hand. The answer should end up as
something that looks something like .
745 divided by 3
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The long division is shown here explicitly to
serve as a refresher for more complicated long division of polynomials.
Therefore, it can be concluded that 745/3 is 248 with aremainder of 1, or, to put it another
way, 745/3 =248 + 1/3.
Long division is a skill that many may have decided they could forget, since calculators
perform this task much faster. However, long division comes roaring back, because here is a
problem that a calculator cannot solve: 6x
3
8x
2
+4x2
2x4
. This problem is solved in a very similar
way as the previous problem.
Start by rewriting the problem in standard long division form (A). Follow along the text with
the graphic .
Polynomial long divion
For explanations of each step, see the text.
Then divide the first time to get 3x2 (B). Why 3x2? This comes from the question: "How many
times does 2x go into 6x3" or, to put the same question another way: "What would I multiply
2x by, in order to get 6x3?" This is comparable to the first step in the long division problem:
"What do I multiply 3 by, to get 7?"
Now (C), multiply the 3x2 times the (2x–4) and get 6x3–12x2. Then subtract this from the line
above it. The 6x3 terms cancel—that shows the right term was picked above. Note that careful
work must be done with the signs, –8x2–(–12x2) gives positive 4x2.
Next, bring down the 4x, as shown in (D). All four steps of long division are now complete—
divide, multiply, subtract, and bring down. At this point, the process begins again, with the
question "How many times does 2x go into 4x2?"
(E) is not the next step. This is merely what the process looks like after all the steps have been
finished. It is a good idea to go through the problem once more to check the work.
Therefore, it can be concluded that put it another way, 3x2 + 2x + 6 +
6x
3
8x
2
+4x2
2x4
22
2x
−4
is 3x
2
+ 2x + 6
with a remainder of 22, or, to
.
If a polynomial can be divided by another equation and have no remainder, then the equation
that was divided by is called afactor. In this case, a remainder will not be written, as the
divisor divided evenly into the dividend.
Be sure to check the answers after doing these types of problems, either by plugging in
numbers, or by multiplying the divisor by the quotient to see if the dividend can be gotten
back!