Digital Lesson
Division of Polynomials
Dividing Polynomials
Long division of polynomials is similar to long division of
whole numbers.
When you divide two polynomials you can check the answer
using the following:
dividend = (quotient • divisor) + remainder
The result is written in the form:
remainder
dividend  divisor  quotient +
divisor
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2
Example: Divide x2 + 3x – 2 by x – 1 and check the answer.
x + 2
x  1 x 2  3x  2
x2 +
x
2x – 2
2x + 2
–4
remainder
Answer: x + 2 +
x2
1. x x   x
x
2. x( x  1)  x 2  x
2
3. ( x 2  3x)  ( x 2  x)  2 x
2x
4. x 2 x 
2
x
5. 2( x  1)  2 x  2
6. (2 x  2)  (2 x  2)  4
–4
x 1
Check: (x + 2) (x + 1) + (– 4) = x2 + 3x – 2 correct
quotient divisor remainder
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dividend
3
Example: Divide 4x + 2x3 – 1 by 2x – 2 and check the answer.
x2 + x + 3
2x  2 2x  0x  4x  1
3
2
2x3 – 2x2
2x2 + 4x
2x2 – 2x
6x – 1
6x – 6
5
Answer: x2 + x + 3 
5
2x  2
Check: (x2 + x + 3)(2x – 2) + 5
= 4x + 2x3 – 1
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Write the terms of the dividend in
descending order.
Since there is no x2 term in the
dividend, add 0x2 as a placeholder.
3
2
x
1.
2. x 2 (2 x  2)  2 x3  2 x 2
 x2
2x
2x2
3
3
2
2
3. 2 x  (2 x  2 x )  2 x
4.
x
2x
5. x(2 x  2)  2 x 2  2 x
6. (2 x 2  4 x)  (2 x 2  2 x)  6 x
8. 3(2 x  2)  6 x  6
7. 6 x  3
2x
9. (6 x  1)  (6 x  6)  5  remainder
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Example: Divide x2 – 5x + 6 by x – 2.
x – 3
x  2 x 2  5x  6
x2 – 2x
– 3x + 6
– 3x + 6
0
Answer: x – 3 with no remainder.
Check: (x – 2)(x – 3) = x2 – 5x + 6
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5
Example: Divide x3 + 3x2 – 2x + 2 by x + 3 and check the answer.
x2 + 0x – 2
x  3 x 3  3x 2  2 x  2
Note: the first subtraction
eliminated two terms from
the dividend.
Therefore, the quotient
skips a term.
Answer:
x2
–2+
8
x3
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x3 + 3x2
0x2 – 2x + 2
– 2x – 6
8
Check: (x + 3)(x2 – 2) + 8
= x3 + 3x2 – 2x + 2
6