Division of Polynomials
To divide a polynomial by a monomial, we can divide each term by the monomial.
Example
4x2 - 2x -
3
4x2 -
2x 3
=
-
-
5x
5x
5x 5x
4x
2 3
=
-
-
5
5 5x
Exercises
Divide the polynomials
-
2x3 + 5x2 - 4
3x
-
7x4 - 3x2 + x - 2
6x2
Recall how to perform long division. We will divide
4321 by 6. We see that we follow the steps:
-
Write it in long division form.
-
Determine what we need to multiply the quotient by to get the first term.
-
Place that number on top of the long division sign.
-
Multiply that number by the quotient and place the product below.
-
Subtract
-
Repeat the process until the degree of the difference is smaller than the degree of the quotient.
-
Write as sum of the top numbers + remainder/quotient.
Example
Divide
4321
6
Solution
720
6 |
4321
42
6 x 7 = 42
12 43 - 42
= 1 and drop down the 2
12 6 x 2
= 12
01 12 - 12 = 0
and drop down the 1
0 6 x 0 =
0
1 1 - 0 =
1
To divide polynomials, we perform the same steps, except that there are x's involved.
Example
Divide
x4
+ 3x2 - 5
x2
+ 4x
Solution
We first write in long division form
Next decide what we need to multiply x2 by to get x4. Since
(x2)(x2) = x4
we can write
Next, we multiply x2 + 7x and x2.
Now subtract to get and bring down the 3x2 to get
We repeat this process until the degree of the remainder is less than the degree of the denominator.
The final solution is
-76x + 5
x2 - 4x + 19
+
x2 + 4x
Exercises
Divide the polynomials.
-
3x2 + 5x + 7
x + 1
-
2x4 + x - 1
x2 + 3x + 1
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