Question

If $x^2+ax+b+6$ has $(x-2)$ as a factors and leaves a reminder $3$ when divided by $(x-3)$ Find $a$ & $b$

Answer 1

$p\left(x\right)=x^2+ax+b+6$ for this $(x-2)$ is a factor $\Rightarrow p\left(2\right)=0$

$p\left(2\right)=0=2^2+2a+b+6$

$0=4+2a+b+6$

$\therefore 2a+b=-10$ ______ (1)

Remainder is $3$ when divided by $(x-3)\Rightarrow p\left(3\right)=3$

$p\left(3\right)=3=3^2+3a+b+6$

$3=a+3a+b+6$

$3a+b=-12$ _____ (2)

Solving (1) and (2)

$a=-2$

$b=-6$