Question

Let f(x) be a cubic polynomial such that the coefficient of $x^3$ is 1. If f(1) = 1, f(2) = 4 and f(3) = 9, then the value of f(5) is

• A. 25
• B. 49
• C. 37
• D. 61

Answer 1

The correct option is B 49

Let the cubic polynomial be $f\left(x\right)=a{x}^3+b{x}^2+cx+d$

Given that a = 1

$\Rightarrow f\left(x\right)=x^3+b{x}^2+cx+d$

Now, $f\left(1\right)=1$ (Given)

$\Rightarrow 1+b+c+d=1$

$\Rightarrow b+c+d=0$ ...(i)

Also f(2) = 4 (Given)

$\Rightarrow (2{)}^3+(2{)}^{2}b+2c+d=4$

$\Rightarrow 8+4b+2c+d=4$

$\Rightarrow 4b+2c+d=-4$ ...(ii)

Also, f(3) = 9 (Given)

$\Rightarrow (3{)}^3+(3{)}^{2}b+3c+d=9$

$\Rightarrow 27+9b+3c+d=9$

$\Rightarrow 9b+3c+d=-18$ ...(iii)

Now, subtracting (i) from (ii), and (i) from (iii), we get



Multiplying (iv) by 2, then subtracting (v) from it, we get


⇒ b = –5

Substitute the value in (iv), we get

3(-5) + c = -4

$\Rightarrow (-15)+c=-4$

$\Rightarrow c=15-4$

$\Rightarrow c=11$

Substituting the values of b and c in (i), we get

$\Rightarrow -5+11+d=0$

$\Rightarrow 6+d=0$

$\Rightarrow d=-6$

Thus, the cubic polynomial is $f\left(x\right)=x^3-5x^2+11x-6.$

Then, $f\left(5\right)=(5{)}^3-5^{\prime }(5{)}^2+{11}^{\prime }5-6$

= 125 - 125 + 55 - 6

= 55 - 6

= 49

Hence, the correct answer is option (b).