Remainder Theorem
Let p(x) be any polynomial of degree n>0 ,and a any real number. If p(x) is divided by ( xa), then the remainder is p(a).
The Remainder Theorem can be proved as follows.
Proof
Let us suppose when p(x) is divided by ( xa), the quotient is q(x) and remainder is r(x).
So we have,
p(x)=(xa) q(x)+r(x), where r(x)=0 or degree of r(x)< degree of xa.
Since degree of ( xa) is1, either r(x)=0 or degree of r(x)=0
So r(x) must be a constant,say r.
Thus for all values of x,
p(x)=(xa) q(x)+r ........(1), where r is a constant.
In particular, when x=a,
p(a)=0.q(x)+r
=0+r
=r
Hence the theorem.
Factor Theorem
Let p(x) be a polynomial of degree n>0. If p(a)=0 for a real number a, then (xa) is a factor of
p(x). Conversely, if (xa) is factor of p(x), then p(a)=0.
Proof
First part:
Let p(a)=0
Then by remander theorem, r=0
So equation (1) becomes
p(x)=(xa) q(x)
==>(xa) is a factor of p(x).
Second Part:
By remainder theorem,
p(x)=(xa) q(x)+r
ie. p(x)=(xa) q(x)+p(a)
Since (xa) is a factor, p(a) must be zero.
This proves the theorem.
Example
Find the remainder when p(x)= x^2 +3x+1 is divided by x+1.
Determine whether( x2) is a factor of p(x) or not.
Solution
x+1= x(1)
[Always check whether the divisor is in the form of (xa)or not. Otherwise rewrite that in the form of(xa)]
So, here a=1
There fore by remainder theorem, the required remainder is
p(a)= p(1)
=(1)^2+3(1)+1
=13+1
=1
We know, by factor theorem,
if (x2) is a factor of p(x), then p(2) must be zero.
Here p(2)=2^2+3(2)+1
=4+6+1
=11 which is not equal to zero.
So (x2) is not a factor of p(x).
Labels: Algebra
1 Comment:

 lala said...
10/27/2007Thank you so much! This really helped me understand the factor theorem. I was having trouble with that.