In
other words 22 goes into 453 with (remainder) 13 left
over.
DIVISION OF
POLYNOMIALS
Objectives: To recall how to divide one polynomial by another.
Recall 1: You may not recall how to use long division to divide 17136 by 36. After all, there is no need to perform such an operation not that calculators are available in any corner store for less than $5.00. However, to understand how to divide one polynomial by another--an operation that no inexpensive calculator can perform (yet)--it would be helpful to review the way long division works. It will turn out that the method of division of polynomials will mimic that of numbers.
476
¨ quotient
divisor-> 36)17156
¨
dividend
144
¨ multiply
first digit of quotient by divisor
273
¨ subtract and bring down a digit
of dividend
252
¨ multiply second digit
of quotient by divisor
216
¨ subtract and bring down a digit of dividend
216
¨ multiply third digit of quotient
by divisor
0
¨ subtract
Recall 2: Dividing polynomials by polynomials of degree 1.
Such division is similar to the long division of real numbers. Let's say that you want to divide x^3 + x^2 - 2x - 8 by (x - 2). The formal procedure is the same.
x^2
+ 3x +
4
¨
quotient
divisor -> x - 2)x^3 + x^2 - 2x +
8
¨
dividend
x^3
-
2x^2
¨
multiply first term of quotient by divisor
3x^2
-
2x
¨
subtract and bring down a term of dividend
3x^2
-
6x
¨
multiply 2-nd term of quotient by divisor
4x
- 8 ¨ subtract,
bring down
term of dividend
4x
- 8 ¨ multiply
3-rd term
of quotient by divisor
0
As you may have noticed, the only difference between dividing numbers and dividing polynomials is that in the first case we do things to digits and in the second we do things to terms.
The process
To divide a polynomial (the dividend) by a polynomial of degree 1 (the divisor) do the following:
1. Arrange both polynomials in descending order of the powers.
2. Divide the first term of the dividend by the first term of the divisor. This will give you the first term of the quotient.
3. Multiply the first term of the quotient by the whole divisor and subtract the result from the dividend.
4. Add the second term of the dividend to the result in step 3.
5. Divide the result in step 4 by the divisor.
6. Continue this process until you cannot divide any longer.
Recall 3: Polynomials are not always written with all powers of x from the highest to the lowest. Always be sure to introduce terms in descending order of the powers of the variable.
For example, if you wish to divide x^3 - 31x + 30 by (x
- 5), be
sure to include the term involving x^2; it may be
written as
0*x^2. So, you should start
with:
x - 5) x^3 + 0x^2 - 31x + 30
If you do not include the missing powers of x, you will find yourself trying to subtract different powers of x to get a power of x. This is not possible. For example, you cannot subtract x^2 from x^3 to get another power of x. In other words, the reason for introducing the extra term 0x^2 is simply to keep the tabulation records straight.
Note
It is important to arrange the
terms in descending
order. If the dividend or the
divisor is
x^2
+ 4x^3 + 4x - 5,
rewrite it as 4x^3 + x^2 + 4x - 5.
Recall
4: If you try to divide 453 by 22,
you will need to
describe the answer with a remainder.
The answer
may be found by long division.
20
22)453
44
13
Since 13 is smaller than 22, the answer is 20
In
other words 22 goes into 453 with (remainder) 13 left
over.
Remainders also happen when dividing polynomials. A remainder of 0 means that the dividend is a product of the divisor and another polynomial. This is not always the case.
For example, try to divide x^2 + 7x - 25 by x - 3.
x
+ 10
x -
3) x^2 + 7x
- 25
x^2
- 3x
10x
- 25
10x
- 30
5
The answer may be expressed as
x + 10 + 5/ (x - 3)
or x + 10 with a remainder of 5 .
Examples
Example 1. Divide x^3 - x^2 - 21x + 36 by x - 4.
Solution:
x^2
+ 3*x - 9
x
- 4)x^3 - x^2 - 21x + 36
¨
dividend
x^3
- 4x^2 ¨ multiply first term
of
quotient by divisor
3x^2
- 21x ¨ subtract and bring down a term of
dividend
3x^2
- 12x ¨ multiply 2-nd term of quotient
by
divisor
-9x
+ 36 ¨ subtract, bring down term of dividend
-9x
+ 36 ¨ multiply 3-rd term of quotient by divisor
0
Example 2. Divide x^5 - 2x^3 - 3x^2 + 6 by x^2 - 2.
Solution: Introduce the intermediate terms in both the divisor and dividend. Even though those terms have the value 0, they will help in the ordering of the columns that must line-up in the calculations.
x^3
- 3
x^2 + 0x - 2 ) x^5 + 0x^4 -
2x^3 - 3x^2 + 0x +
6
¨ dividend
x^5
+ 0x^4 -
2x^3
¨
multiply first term of quotient by divisor
0x^3
- 3x^2 + 0x + 6 ¨ subtract and bring down three
-3x^2 + 0x +
6
more
terms
0
In this example it was necessary to bring down several
terms
of the dividend at a time. This is done for exactly the same
reason
that we bring down more than one term in the long division of 4515
by
22. In that case, we get
205
22 )4515
44
110
¨ subtract and bring down two terms
110
0
Example 3. Divide 6x^3 + 14x + 17x^2 + 3 by 2x + 3.
Solution: First, rearrange the terms so that they are in descending order of the powers of x. That means rewrite the dividend as 6x^3 + 17x^2 + 14x + 3.
3x^2
+ 4x + 1
2x + 3 )
6x^3 + 17x^2 + 14x
+ 3 ¨
dividend
6x^3
+ 9x^2
8x^2
+ 14x
8x^2
+ 12x
2x
+ 3
2x
+ 3
0
Example 4. Divide x^3 + 10x^2 + 12x + 1 by x + 3.
Solution: This is a case where the divisor does not divide completely into the dividend.
x^2
+ 7x - 9
x + 3 ) x^3
+ 10x^2 +
12x + 1 ¨
dividend
x^3
+ 3x^2
7x^2
+ 12x
7x^2
+ 21x
-9x
+ 1
-9x
- 27
28 ¨
remainder
So, the answer is x^2 + 7x - 9
with a remainder
of 28 .
Note
In Example 3 we had the situation where the first term of the dividend was a nice multiple of the divisor; that is, 6x^3 is a multiple of 2x. We have not given an example of a situation where fractional coefficients arise; but, that does not mean to say that such a situation does not occur. What would you do if you were asked to divide x^3 + 14x + 17x^2 + 3 by 2x + 3? A little thought reveals that the first term of your quotient must be 1/2 x^2 Dealing with fractions can be a bit unnerving, so we shall not continue with such an example at this time. The important point is that you should be familiar with the method of long division of fractions.
EXERCISES
1. Divide x^2 - x - 2
by x - 2.
2.
Divide x^4 + 2x^3 + x^2 by x +
1
3 Divide x^3 -
x^2 - x + 1 by x^2 -
1
4 Divide 3x^3 + 5x^2 + 2x by 3x +
2
5 Divide 16x^4 - 1
by 2x - 1
6 Divide
16x^4 - 1 by 4x^2 + 1
7 Divide x^5 - 2x^4 - x^3 + 2x^2 + x - 2 by
x -
2
8 Divide x^4 - x^3 + 5x^2
- 5x by x^2 + 5.
9 Divide 8x^4 + 10x^2 - 3 by 2x^2 +
3.
10. Divide x^4 -
2x^3 + 2x^2 - 2x + 1
by x^2 - 2x +
1.
11
Divide x^4 + 13x^2 + 36 by x^2 -x -
6
12 Divide 4x^4 -
25x^2 + 36 by 2x^2 - x -
6
13 Divide x^4 + 2x^3 + x^2 + 3 by x +
1
14
Divide x^3 - x^2 - 2x + 3 by
x^2 -
1
15 Divide 4x^4 - 25x^2 + 20 by 2x^2 - x - 6