Long and synthetic devision of polynomial

1
Division of higher degree Polynomials
2
Dividing 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? division
algorithm ?
3
Example Divide Check
Example Divide x2 3x 2 by x 1 and check
the answer.
x
2
x2 x
2x
2
2x 2
4
remainder
Check
correct
4
Example Divide Check
Example Divide 4x 2x3 1 by 2x 2 and check
the answer.
x2
x
3
Write the terms of the dividend in descending
order.
2x3 2x2
Since there is no x2 term in the dividend, add
0x2 as a placeholder.
2x2
4x
2x2 2x
6x
1
6x 6
5
Check (x2 x 3)(2x 2) 5
4x 2x3 1
5
Example Division With Zero Remainder
Example Divide x2 5x 6 by x 2.
x
3
x2 2x
3x
6
3x 6
0
Answer x 3 with no remainder.
Check (x 2)(x 3)
x2 5x 6
6
Example Division With Nonzero Remainder
Example Divide x3 3x2 2x 2 by x 3 and
check the answer.
x2
2
0x
Note the first subtraction eliminated two terms
from the dividend.
x3 3x2
0x2
2x
2
2x 6
Therefore, the quotient skips a term.
8
Check (x 3)(x2 2) 8
x3 3x2 2x 2
7
Synthetic Division
8
Ruffinis rule3 or Synthetic Division

• Paolo Ruffini (September 22, 1765 May 10, 1822)
  was an Italian mathematician and philosopher.
• By 1788 he had earned university degrees in
  philosophy, medicine/surgery, and mathematics.
  Among his work was an incomplete proof
  (AbelRuffini theorem1) that quintic (and
  higher-order) equations cannot be solved by
  radicals (1799), and Ruffini's rule3 which is a
  quick method for polynomial division.

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Synthetic Division
There is a shortcut for long division as long as
the divisor is x k where k is some number.
(Can't have any powers on x).
1
- 3
1 6 8 -2
- 3
Add these up
- 9
3
Add these up
Add these up
1
3
- 1
1
x2 x
This is the remainder
Put variables back in (one x was divided out in
process so first number is one less power than
original problem).
List all coefficients (numbers in front of x's)
and the constant along the top. If a term is
missing, put in a 0.
So the answer is
10
Let's try another Synthetic Division
0 x3
0 x
1
4
1 0 - 4 0 6
4
Add these up
16
48
192
Add these up
Add these up
Add these up
1
4
12
48
This is the remainder
x3 x2 x
198
Now put variables back in (remember one x was
divided out in process so first number is one
less power than original problem so x3).
List all coefficients (numbers in front of x's)
and the constant along the top. Don't forget the
0's for missing terms.
So the answer is
11
Let's try a problem where we factor the
polynomial completely given one of its factors.
You want to divide the factor into the polynomial
so set divisor 0 and solve for first number.
- 2
4 8 -25 -50
- 8
Add these up
0
50
Add these up
Add these up
No remainder so x 2 IS a factor because it
divided in evenly
4
0
- 25
0
x2 x
Put variables back in (one x was divided out in
process so first number is one less power than
original problem).
List all coefficients (numbers in front of x's)
and the constant along the top. If a term is
missing, put in a 0.
So the answer is the divisor times the quotient
You could check this by multiplying them out and
getting original polynomial
12
EXAMPLE 1
Use synthetic division
Divide f (x) 2x3 x2 8x 5 by x 3 using
synthetic division.
SOLUTION
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