If the polynomial f(x) = $\displaystyle ax^3 + bx - c$ is divisible by the polynomial g(x) = $\displaystyle x^2 + bx +c$, then $\displaystyle ab = $ ?
Please Help....
If the hypothesis id true is...
$\displaystyle a\ x^{3} + b\ x -c = (a\ x + d)\ ( x^{2} + b\ x + c)$ (1)
... and that means that is...
$\displaystyle a\ b + d=0$
$\displaystyle c\ d = -c$ (2)
... so that...
$\displaystyle d=-1$
$\displaystyle a\ b=1$ (3)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Now...
a) if $\displaystyle x^{2} + b\ x + c$ devides $\displaystyle a\ x^{3} + b\ x -c$ , then there is a first order polynomial $\displaystyle a\ x + d$ so that...
$\displaystyle a\ x^{3} + b\ x -c = (a\ x + d)\ (x^{2} + b\ x + c) $ (1)
b) if You develop the product (1) You obtain...
$\displaystyle (a\ x + d)\ (x^{2} + b\ x + c) = a\ x^{3} + (a\ b + d)\ x^{2} + (a\ c + b\ d) x + d\ c = a\ x^{3} + b\ x -c $ (2)
c) from the (2) You obtain ...
$\displaystyle a\ b + d=0$ , $\displaystyle d= -1 \rightarrow a\ b = 1$ (3)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$