Question

f(x) = x^3 - ax^2 - bx - 3a and g(x) = x^3 -(a-2)x^2 - bx -3b have same quadratic factor. How to find a and b value?

Answer 1

since they have a common quadratic factor, there will be two common roots as well :
say, \(p, q, r\) are roots of \(f(x)\) and \(p,q,s\) are roots of \(g(x)\)

\(f(x) = x^3 - ax^2 - bx - 3a = (x-p)(x-q)(x-r)\)
\(g(x) = x^3 -(a-2)x^2 - bx -3b =(x-p)(x-q)(x-s)\)

Answer 2

equation \(x\) coefficient in both equations, gives :-
\(-b = pq + qr + rp\)
\(-b = pq + qs + sp\)

Answer 3

subtracting both :-

\(0 = q(r-s)+ p(r-s) = (r-s)(q+p)\)
for this to be true, atleast one factor must equal 0
\(r=s, \) or \(p = -q\)

Answer 4

If r = s, f(x) and g(x) will be one and the same function.
So p = -q.

Answer 5

Oh ya, so we can discard r=s
and pursue p = -q case

Answer 6

still its not fully done...
u sure u can work the rest ha ? :)

Answer 7

i see lot of algebra still.... good luck !

Answer 8

this is my lil' sister homework, she is in high school, but i don't know why its so difficult :(

Answer 9

she must be in12th grade ?

Answer 10

There are 6 unknowns: a, b, p, q, r, s
and there are 6 equations, equating the coefficients of x^2, x and constant in to the two functions coefficients.

Answer 11

I'm stuck -__-