Unformatted text preview: can be found by dividing P ( x ) by x-r .
There is one more fact that we need to get out of the way.
If P ( x ) = ( x - r ) Q ( x ) and x = t is a zero of Q ( x ) then x = t will also be a zero of P ( x ) .
This fact is easy enough to verify directly. First, if x = t is a zero of Q ( x ) then we know that, Q (t ) = 0
since that is what it means to be a zero. So, if x = t is to be a zero of P ( x ) then all we need to
do is show that P ( t ) = 0 and that’s actually quite simple. Here it is, P (t ) = (t - r ) Q (t ) = (t - r ) ( 0) = 0
and so x = t is a zero of P ( x ) .
Let’s work an example to see how these last few facts can be of use to us. Example 3 Given that x = 2 is a zero of P ( x ) = x3 + 2 x 2 - 5 x - 6 find the other two zeroes.
First, notice that we really can say the other two since we know that this is a third degree
polynomial and so by The Fundamental Theorem of Algebra we will have exactly 3 zeroes, with
some repeats possible.
So, since we know that x = 2 is a zero of P...
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This note was uploaded on 06/06/2012 for the course ICT 4 taught by Professor Mrvinh during the Spring '12 term at Hanoi University of Technology.
- Spring '12