DIVIDE POLYNOMIALS
Divide a Polynomial by a Monomial:
Divide each term of the polynomial by the monomial.
Examples:
1) (4a3 - 6a2 - 12a) ÷ (2a)
4a3 - 6a2 - 12a
2a
2a
2a
=
2a2 - 3a - 6
(recall quotient rule)
2) 36m3n3 + 15m4n2 + 3m5n
3m5n
36m3n3 + 15m4n2 + 3m5n
3m5n
3m5n
3m5n
12m-2n2 + 5m-1n + 1 = 12n2 + 5n + 1
m2
m
(recall negative power rule)
Divide a Polynomial by a Polynomial:
The solution is found by performing long division. The following are the steps.
Recall how to set up long division and the terms:
(6x2 + 19x + 13) ÷ (2x + 5)
2x + 5 is the divisor
3x + 2 +
3
2x+5
or
6 x 2  19 x  13
2x  5
or
2x+5 6 x 2  19 x  13
6x2 + 19x + 13 is the dividend
is the quotient (answer) with a remainder
STEPS:
1) Put both polynomials in standard form (descending powers of the variable).
2) Divide the first term of the dividend by the first term of the divisor, and write the result over
the dividend.
3) Multiply the result in step 2 by each term in the divisor, and write the result under the
dividend.
4) Subtract the result in step 3 from the dividend. (or take the opposite sign of result in step 3)
5) Bring the next term down.
6) Repeat step 2 through step 5 until there are no other terms to bring down. If you have a
remainder, then place it over the divisor.
Examples:
1)
____________
2x + 5 | 6x2 + 19x + 13
given expression
3x
2x + 5 | 6x2 + 19x + 13
6x2 + 15x
step 2: 6x2 ÷ 2x = 3x
step 3: 3x(2x + 5)
3x
2x + 5 | 6x2 + 19x + 13
-6x2 - 15x
+ 4x
step 4: -(6x2 + 15x) and collect like terms
3x
2x + 5 | 6x2 + 19x + 13
-6x2 - 15x
+ 4x + 13
step 5: bring down next term
3x + 2
2x + 5 | 6x2 + 19x + 13
-6x2 - 15x
+ 4x + 13
4x + 10
3x + 2
2x + 5 | 6x2 + 19x + 13
-6x2 - 15x
+ 4x + 13
- 4x - 10
+3
repeat steps 2-5
step 2: 4x ÷ 2x = 2
step 3: 2(2x + 5)
step 4: -(4x + 10) and collect like terms.
There are no more terms to bring down.
Since there is a remainder, write it over the divisor.
The final answer is 3x + 2 + 3 .
2x + 5
2)
______________
3x + 2 | 3x3 - x2 - 11x - 8
given expression
x2
3x + 2 | 3x3 - x2 - 11x - 8
3x3 + 2x2
step 2: 3x3 ÷ 3x = x2
step 3: x2(3x + 2)
x2
3x + 2 | 3x3 - x2 - 11x - 8
-3x3 - 2x2
- 3x2
x2
3x + 2 | 3x3 - x2 - 11x - 8
-3x3 - 2x2
-3x2 - 11x
x2 - x
3x + 2 | 3x3 - x2 - 11x - 8
-3x3 - 2x2
-3x2 - 11x
-3x2 - 2x
x2 - x
3x + 2 | 3x3 - x2 - 11x - 8
-3x3 - 2x2
-3x2 - 11x
+3x2 + 2x
-9x - 8
x2 - x - 3
3x + 2 | 3x3 - x2 - 11x - 8
-3x3 - 2x2
-3x2 - 11x
+3x2 + 2x
-9x - 8
-9x - 6
x2 - x - 3
3x + 2 | 3x3 - x2 - 11x - 8
-3x3 - 2x2
-3x2 - 11x
+3x2 + 2x
-9x - 8
+9x + 6
-2
Final answer is x2 - x - 3 -
2 .
3x+2
step 4: -(3x3 + 2x2) and collect like terms
step 5: bring down the next term
repeat steps 2-5
step 2: -3x2 ÷ 3x = -x
step 3: -x(3x + 2)
step 4: -(-3x2 - 2x )
step 5: bring down next term
repeat steps 2-5
step 2: -9x ÷ 3x = -3
step 3: -3(3x + 2)
step 4: -(-9x - 6)
There are no more terms to bring down.
You have a remainder.
3) (x4 + 25) ÷ (x + 5)
given expression
Note: the dividend, x4 + 25, is missing the terms with power of 3, 2, and 1. We need to include
them into the dividend by placing 0x3, 0x2, and 0x in descending order. These are considered
place holders. It is important to recognize that the term must have the coefficient of zero. This is
because we don't want to change the original value of the problem. It is similar to the following
concept, 4+5 = 4+0+0+5 = 9.
_____________________
x + 5 | x4 + 0x3 + 0x2 + 0x + 25
_x3 +_______________
x + 5 | x4 + 0x3 + 0x2 + 0x + 25
x4 + 5x3
step 1: set up as long division
with place holders
step 2: x4 ÷ x = x3
step 3: x3(x + 5)
__x3 +_______________
x + 5 | x4 + 0x3 + 0x2 + 0x + 25
-x4 - 5x3
-5x3 + 0x2
step 4: -(x4 + 5x3)
step 5: bring down next term
_x3 - 5x2______________
x + 5 | x4 + 0x3 + 0x2 + 0x + 25
-x4 - 5x3
-5x3 + 0x2
-5x3 -25x2
step 2: -5x3 ÷ x = -5x2
step 3: -5x2(x + 5)
_x3 - 5x2______________
x + 5 | x4 + 0x3 + 0x2 + 0x + 25
-x4 - 5x3
-5x3 + 0x2
+5x3 +25x2
+25x2 + 0x
_x3 - 5x2_+ 25x _____
x + 5 | x4 + 0x3 + 0x2 + 0x + 25
-x4 - 5x3
-5x3 + 0x2
+5x3 +25x2
+25x2 + 0x
+25x2 + 125x
_x3 - 5x2_+ 25x_______
x + 5 | x4 + 0x3 + 0x2 + 0x + 25
-x4 - 5x3
-5x3 + 0x2
+5x3 +25x2
+25x2 + 0x
-25x2 - 125x
-125x + 25
step 4: -(-5x3 -25x2)
step 5: bring down next term
repeat steps 2-5
step 2: +25x2 ÷ x = +25x
step 3: 25x(x + 5)
step 4: -(+25x2 + 125x)
step 5: bring down next term
_x3 +_-5x2_+ 25x - 125______
x + 5 | x4 + 0x3 + 0x2 + 0x + 25
-x4 - 5x3
-5x3 + 0x2
+5x3 +25x2
+25x2 + 0x
-25x2 - 125x
-125x + 25
-125x - 625
_x3 - 5x2_+ 25x - 125______
x + 5 | x4 + 0x3 + 0x2 + 0x + 25
-x4 - 5x3
-5x3 + 0x2
+5x3 +25x2
+25x2 + 0x
-25x2 - 125x
-125x + 25
+125x + 625
+650
The final answer is x3 -5x2 + 25x - 125 + 650 .
x+5
step 2: -125x ÷ x = -125
step 3: -125(x + 5)
step 4: -(-125x - 625)
There are no more terms to bring down.