• Best of luck to the class of 2025 for their HSC exams. You got this!
    Let us know your thoughts on the HSC exams here

Polynomial help! (1 Viewer)

EpikHigh

Arizona Tears
Joined
Dec 17, 2011
Messages
499
Gender
Male
HSC
2013
Find a and b if x^4 -2x^3 + x^2 + ax + b is exactly divisble by x^2 + x - 1

Thanks!
 

Sy123

This too shall pass
Joined
Nov 6, 2011
Messages
3,725
Gender
Male
HSC
2013
Express the polynomial in terms of its quotient, remainder and divisor:



For it to be exactly divisible, R(x)=0



Notice how A(x) will have to be some quadratic equation:



Note how since x^4 has co-efficient of 1, hence p must be 1, since x^2 * px^2 = x^4.



Group them up by their respective co-efficients



Equate co-efficients:

q + 1 = -2
q = -3

r + q -1 = 1

r - 3 - 1 = 1

r = 5

r - q = a

5 - 3 = 2 = a

Therefore a = 2

-r = b

Therefore b = -5

========


I can see another method by subbing in the roots of the quadratic x^2+x-1, but that just looks too tedious.
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top