Polynomial help! (2 Viewers)

[EpikHigh]

Find a and b if x^4 -2x^3 + x^2 + ax + b is exactly divisble by x^2 + x - 1

Thanks!

 

[Sy123]

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

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

 

[Carrotsticks]

Find a and b if x^4 -2x^3 + x^2 + ax + b is exactly divisble by x^2 + x - 1

Thanks!