MHF4U
March 18 2013
3.5 Long Division of Polynomials:
Remainder
Quotient
Divisor
The degree of the remainder is strictly less than the degree of the divisor.
We can rearrange this expression as rational functions to show the division:
Ex: Evaluate , using long division.
Multiply (x+1) by x to get x2+x
Multiply (x+1) by 1to get x+1
MHF4U
March 18 2013
MHF4U
March 18 2013
MHF4U
March 18 2013
3.6 Factoring:
In order to factor polynomials we would like to have two things:
a) A method to identify possible factors
b) A means to quickly test whether a candidate is an actual factor.
Rational Root Theorem
Consider a polynomial in factored form:
If
is a factor of
, then p is a divisor of and q is a divisor of
This means that we can find all possible rational roots, by looking at all the divisors of the a0 and an.
Ex: Identify all the possible rational roots of
Solution: The roots will be of the form
where p divides 3 and q divides 6
This still leaves us with the problem of how to quickly identify which of these are the actual roots.
MHF4U
March 18 2013
MHF4U
March 18 2013
Remainder Theorem:
Recall from 3.5 that:
Since the degree of r(x) must be less than the degree of (x­a), r(x) must be a constant.
The remainder of is a factor of
Ex: Determine whether Solution:
is
.
, if and only if or .
divides
The remainder is 0 so divides
The remainder is ­3 so does not divide
MHF4U
March 18 2013
MHF4U
March 18 2013
f(x)
Common Factor
q(x)
Is the function a quadratic?
Yes
Use Quadratic Equation
No
Determine all possible rational roots using rational root theorem
q(x)
Test the rational roots using the remainder theorem, to find a factor
Use Long Division to find quotient q(x)
q(x)
MHF4U
March 18 2013