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Polynomial Division
Division Long and Short
Polynomial Division 28/10/2013
Division of Polynomials Long Division (by hand):
Polynomial Functions
245 ) 98154
98015
0 Quotient
Remainder
9815245
We just did !
So, = 40 + 349
Can we do this with polynomials ?
= 40 +15245
*
*See Notes page
Polynomial Division 38/10/2013
General Division of Polynomials For polynomials f(x) and d(x) , d(x) ≠ 0
where Q(x) is the quotient , r(x) is the remainder , d(x) is the divisor and f(x) is the dividend
Thus
where either r(x) = 0 or deg r(x) < deg d(x)
Polynomial Functions
+f(x)d(x) = Q(x) r(x)
d(x)
f(x) = d(x) ∙ Q(x) + r(x)
Polynomial Division 48/10/2013
Algebraic Monomial Division Example:
Polynomial Functions
12x4 + 27x3 – 9x2 + 6x – 2 3x2
Quotient
Remainder
= 12x4
3x2 + – + –9x2
3x2
6x3x2
27x3
3x2
23x2
23x2
–2x+= 4x2 + 9x 3 –
3x22–6x+= 4x2 + 9x 3 –
Polynomial Division 58/10/2013
Arithmetic Monomial Division Example
Polynomial Functions
12x4 + 27x3 – 9x2 + 6x – 2 3x2 )4x2
12x4
27x3
+ 9x
27x3
– 9x2
– 3
– 9x2
6x – 2
Remainder
Quotient
Note: deg r(x) = 1 < deg d(x) = 2
Polynomial Division 68/10/2013
Division by Linear Binomials Example
Polynomial Functions
x4 + 3x3 – 4x + 1 x + 2x3
x4 + 2x3
x3
+ x2
x3 + 2x2
– 2x2
– 2x
– 2x2 – 4x 1
– 4x
)Quotient
Remainder
Question:
Can we do this faster or more simply ?
Polynomial Division 78/10/2013
Arithmetic operations involve only the coefficients
Synthetic Division
2x4 – 3x3 + 5x2 + 4x + 3 x + 2 )2x3
2x4 + 4x3
–7x3
– 7x2
–7x3 – 14x2
19x2
– 34
19x2 + 38x – 34x
+ 4x
+19x
+ 5x2
+ 3– 34x – 68
71Remainder
Example:
NOTE: deg r(x) = 0 < 1 = deg d(x)
Polynomial Division 88/10/2013
Using synthetic division we deal only with the coefficients
Synthetic Division
2x4 – 3x3 + 5x2 + 4x + 3 x + 2 ) Example:
2 –3 5 4 32
24
–7 –14 19
38 –34
–68 71
Subtract
d(x) = x + 2
Remainder–2
2–4–7
14 19
–38 –34
68 71
Addd(x) = x – (–2) 2 –3 5 4 3
Polynomial Division 98/10/2013
Degree Facts For any polynomials A(x), B(x)
deg (A(x) • B(x))
deg (A(x) + B(x))
Examples: deg ( (x2 + 1) • (3x3 – 4x2 + 5x + 7) )
deg ( (3x2 – 4) + (2x4 + 6x + 3) )
Polynomial Functions
= deg A(x) + deg B(x)
= max { deg A(x), deg B(x) }
= 2 + 3 = 5
= max { 2, 4 } = 4
Polynomial Division 108/10/2013
Division Algorithm for Polynomials
Consider polynomial functions f(x), d(x) with
0 < deg d(x) < deg f(x)
There exist unique polynomial functions
Q(x) and r(x) such that
f(x) = d(x) • Q(x) + r(x)
where either r(x) = 0 or deg r(x) < deg d(x)
Polynomial Functions
Polynomial Division 118/10/2013
Division Algorithm
f(x) = d(x) • Q(x) + r(x)
with r(x) = 0 or deg r(x) < deg d(x)
Polynomial Functions
(continued)
Note: This just says that f(x)d(x) Q(x)= +
r(x)d(x)
Polynomial Division 128/10/2013
Division Algorithm
Polynomial Functions
(continued)
Example:
x2 – 1 x – 1
= x + 1deg f(x) = 2
deg d(x) = 1 , r(x) = 0
So,f(x) = d(x) • Q(x) + r(x)
x2 – 1 = (x – 1) • (x + 1) + 0
f(x)d(x) Q(x)= +
r(x)d(x)
becomes
Polynomial Division 138/10/2013
Division Algorithm and Degrees Given
f(x) = d(x) • Q(x) + r(x)
where
either r(x) = 0 or deg r(x) < deg d(x)
Polynomial Functions
Polynomial Division 148/10/2013
Division Algorithm and Degrees Given
f(x) = d(x) • Q(x) + r(x) Question: Suppose
deg d(x) = m > 0 ,
deg Q(x) = n ,
deg f(x) = p > m
What is the relationship among m, n and p ?
Polynomial Functions
Polynomial Division 158/10/2013
Division Algorithm and Degrees
f(x) = d(x) • Q(x) + r(x)
Polynomial Functions
p = deg f(x)
= deg ( d(x) • Q(x) + r(x) )
= max { deg (d(x) • Q(x)) , deg r(x) }
= max { (deg d(x) + deg Q(x)) , deg r(x) }
Polynomial Division 168/10/2013
Division Algorithm and Degrees
Polynomial Functions
= m + n
Since deg r(x) < deg d(x)
then
p = deg f(x)
= max { (deg d(x) + deg Q(x)) , deg r(x) }
= max { (m + n) , deg r(x) }
= m< m + n
max { (m + n) , deg r(x) }
Polynomial Division 178/10/2013
Division Algorithm and Degrees
f(x) = d(x) • Q(x) + r(x)
Polynomial Functions
= m + n
p = deg f(x)
= max { (m + n) , deg r(x) }
p = m + nSo
Polynomial Division 188/10/2013
Think about it !
