Question

We want to find a polynomial f(x) of degree n such that f(1) = $\sqrt{2}$ and f(3) =$\pi$. Which of the following is true?

• A. There does not exist such a polynomial
• B. There is exactly one such polynomial and it has degree 1
• C. There are infinitely many such polynomials for each n $\ge$1
• D. There are infinitely many such polynomials for each n $\ge$ 2 but not infinitely many for n =1

Answer 1

The correct option is D There are infinitely many such polynomials for each n $\ge$ 2 but not infinitely many for n =1

f(x) $\to$ degree(n)
$f\left(1\right)=f\sqrt{2}$
$f\left(3\right)=\pi$
$f\left(x\right)=a{x}^n+b{x}^{n-1}+c{x}^{n-2}$ .......
If degree 0 $f\left(x\right)=a=constant$
$f\left(1\right)=f\left(3\right)$ but it is not true
If degree1 $f\left(x\right)=ax+b$
$f\left(1\right)=a+b=\sqrt{2}$
$f\left(3\right)=3a+b=\pi$
Two variables & two unknown
a & b can be found uniquly
$\therefore$ one polynomial used only
For n > 1
Let n = 2 ax${}^2$ + bx + c
We have 3 variable & only 2 equations can be formed from given condition
Hence infinite such polynomial can be formed.