|
Question 64642: I am struggling with some problems. Could someone please help me!!!!
1. Find a third degree polynomial with real coefficents and with real zeros 2 and 4+i. Write your answer in ax^3+bx^2+cx+d form. I have x^3-10x^2+33x-34 is that correct and is there anything else I should do to it?
2. Use Decartes' Rule of Sign to find the number of possible number of positive zeros and the possible number of negative zeros of Q(x)= x^4-2x^3+4x^2-3x-2. I am completely stuck on this one and don't even know where to start.
Thank you for your help!!!!
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! I am struggling with some problems. Could someone please help me!!!!
1. Find a third degree polynomial with real coefficents and with real zeros 2 and 4+i. Write your answer in ax^3+bx^2+cx+d form. I have x^3-10x^2+33x-34 is that correct and is there anything else I should do to it?
SINCE COEFFICIENTS ARE REAL ,THEN COMPLEX ROOTS SHOULD BE IN CONJUGATES.
HENCE IF 4+i IS A ROOT,THEN 4-i SHOULD ALSO BE A ROOT.
HENCE THE 3 ROOTS ARE 2,4+i,4-i.HENCE POLYNOMIAL IS
(X-2)[X-(4+i)][X-(4-i)]=0
(X-2)[(X-4)^2-i^2]=0
(X-2)[X^2-8X+16+1]=0
X^3 - 8X^2+17X-2X^2+16X-34=0
X^3-10X^2+33X-34=0...........OK
2. Use Decartes' Rule of Sign to find the number of possible number of positive zeros and the possible number of negative zeros of Q(x)= x^4-2x^3+4x^2-3x-2. I am completely stuck on this one and don't even know where to start.
THE RULE SAYS.FIND CHANGES IN SIGN OF COEFFICIENTS OF THE TERMS OF THE FUNCTION WHEN WRITTEN IN A DESCENDING ORDER.
WE ALREADY HAVE THE FUNCTION IN DECENDING ORDER.
WE HAVE HERE ONE CHANGE FROM +1 TO -2
2 ND CHANGE FROM -2 TO +4
3 RD. CHANGE FROM +4 TO -3
SO THE NUMBER OF CHANGES =3 = MAXIMUM NUMBER OF POSITIVE RATIONAL ROOTS.THEY COULD BE 3 OR 3-2 = 1.
Thank you for your help!!!!
|
|
|
| |
