Determine the two zeroes - factor theorem, Algebra

Given that x=2 is a zero of P ( x ) = x3 + 2x2 - 5x - 6 determine the other two zeroes.

Solution

Firstly, notice that we actually can say the other two since we know that it is a third degree polynomial and thus by The Fundamental Theorem of Algebra we will contain exactly 3 zeroes, with some repeats possible.

Thus, since we know that can write P (x) as, x=2 is a zero of P ( x ) = x3 + 2 x2 - 5x - 6 the Fact 1 tells us that we

                                                P (x) =(x - 2) Q (x)

and Q ( x ) will be a quadratic polynomial. Then we can determine the zeroes of Q (x) by any of the methods which we've looked at to this point & by Fact 2 we know that the two zeroes we obtain from Q ( x ) will also by zeroes of P ( x ) .  At this point we'll contain 3 zeroes and thus we will be done.

Hence, let's find Q (x) .  To do this all we have to do is a quick synthetic division as follows.

1205_Determine the two zeroes - Factor Theorem.png

Before writing down Q ( x ) remember that the final number in the third row is the remainder and that we know that P ( 2) have to be equal to this number.  Thus, in this case we have that P ( 2) = 0 .  If you think regarding it, we have to already know this to be true. We were given into the problem statement the fact that x= 2 is a zero of P (x) and that means that we ought to have P ( 2) = 0 .

Thus, why go on regarding this? It is a great check of our synthetic division.  As we know that x= 2 is a zero of P ( x ) and we obtain any other number than zero in that last entry we will know that we've done something incorrect and we can go back and determine the mistake.

Now, let's get back to the problem.  From the synthetic division,

                                     P (x) =(x - 2) ( x2 + 4 x + 3)

Thus, this means that,

Q (x) = x2 + + 4 x + 3

and we can determine the zeroes of this. Here they are,

Q ( x )= x2 + 4 x + 3 = ( x + 3) ( x + 1)

⇒         x= -3, x = -1

Thus, the three zeroes of P ( x ) are x= -3 , x= -1 & x=2 .

As an aside to the earlier example notice that now we can also completely factor the polynomial get,

                                  P ( x ) = x3 + 2 x - 5x - 6 . 

Substituting the factored form of Q ( x ) into P ( x ) we

                             P (x ) = ( x - 2) ( x + 3) (x + 1)

Posted Date: 4/8/2013 2:42:35 AM | Location : United States







Related Discussions:- Determine the two zeroes - factor theorem, Assignment Help, Ask Question on Determine the two zeroes - factor theorem, Get Answer, Expert's Help, Determine the two zeroes - factor theorem Discussions

Write discussion on Determine the two zeroes - factor theorem
Your posts are moderated
Related Questions
what are the solutions of [x+6]_> 5? write the solution as either the union or the intersection of two sets


Interval notation The next topic that we have to discuss is the idea of interval notation.  Interval notation is some very pleasant shorthand for inequalities & will be utilize


A blue whale gains 2.3 tons a month for the first year of life. If a blue whale weighed 15 tons at birth, how much will it weigh on its first birthday?



Find out the partial fraction decomposition of each of the following. 8x 2 -12/( x( x 2 + 2 x - 6) Solution In this case the x which sits in the front is a linear term


Mr. Rodriquez comments on a recent test. "well, the class average on the test is exactly 83. if I take away the best score, the average is exactly 82." if there are 16 students in