Question

If $x^4-x^3+a{x}^2+x+b$ is exactly divisible by $(x-2)$ as well as $(x-3)$ then______.

• A. $a+b=1$
• B. $a+2b=5$
• C. $a=b+1$
• D. all the above
• E. none of above

Answer 1

Answer: (E)

none of above

$Given$
$x^4-x^3+a{x}^2+x+bisexactlydivisiblebyx-2$
$whichmeanx=2isafactorhence$
${\left(2\right)}^4-{\left(2\right)}^3+a{\left(2\right)}^2+2+b=0$
$\Rightarrow 16-8+4a+2+b=0$
$\Rightarrow 10+4a+b=0.........\left(i\right)$
$x^4-x^3+a{x}^2+x+bisexactlydivisiblebyx-3$
$whichmeanx=3isafactorhence$
${\left(3\right)}^4-{\left(3\right)}^3+a{\left(3\right)}^2+3+b=0$
$\Rightarrow 81-27+9a+3+b=0$
$\Rightarrow 57+9a+b=0........\left(ii\right)$
$subtracting\left(i\right)from\left(ii\right)$
$a=\frac{47}{5},b=\frac{-238}{5}$