Question

If $f\left(x\right)$ is polynomial of degree $3$ with leading coefficient $2$ and $f\left(1\right)=4,f\left(2\right)=7,f\left(3\right)=12$ and $f\left(5\right)=11\alpha -1,$ then $\alpha =$

• A. $11$
• B. $10$
• C. $7$
• D. $5$

Answer 1

The correct option is C $7$

$f\left(1\right)=4,f\left(2\right)=7,f\left(3\right)=12$
By the given values, we can assume
$f\left(x\right)=x^2+3$ for $x=1,2,3$
Now, let $g\left(x\right)=f\left(x\right)-x^2-3$, then
$x=1,2,3$ are roots of $g\left(x\right)$
As $f\left(x\right)$ is a polynomial of degree $3$, so degree of $g\left(x\right)$ is also $3$
Therefore,
$g\left(x\right)=a(x-1)(x-2)(x-3)\Rightarrow f\left(x\right)=a(x-1)(x-2)(x-3)+x^2+3$
As the leading coefficient of $f\left(x\right)$ is $2$, so
$f\left(x\right)=2(x-1)(x-2)(x-3)+x^2+3\Rightarrow f\left(5\right)=2\left(4\right)\left(3\right)\left(2\right)+25+3\Rightarrow 11\alpha -1=76\therefore \alpha =7$