Question

Find the values of $a$ and $b$ where $x-2$ and $x+3$ are the factors of equation $x^3+a{x}^2+bx-12=0$.

Answer 1

$x^3+a{x}^2+bx-12=0$ ... (1)

now roots of this equations are $x-2=0$ & $x+3=0$

i.e. (2, -3)

Let $\lambda ,\beta ,\gamma$ be roots of eq. (1) then

$\lambda +\beta +\gamma =-a$...(2)

$\lambda \beta +\beta \gamma +\gamma \lambda =b$....(3)

$\lambda \beta \gamma =+12$...(4)

From eq (4) $\gamma =\frac{12}{+2\times (-3)}=-2[\lambda =2,\beta =-3]$

now from eq. (2) $a=-(2-3-2)=3$

from eq. (3) $b=-6+6-4=-4$

Hence $[a=3]$ and $[b=-4]$