Poly functional

Algebra Level 4

$\begin{cases} {f(1) = 1} \\ {f(2) = 8 } \\ {f(3) = 27} \\ { f(4) = 64} \\ {f(5) =127} \end{cases}$

If $f(x)$ is polynomial having degree 4 that satisfy the system of equations above, what is the value of $f(6)$ ?

The answer is 226.

2 solutions

Shivamani Patil
May 25, 2015

Define $g\left( x \right) =f\left( x \right) -{ x }^{ 3 }$ .

$\therefore g(1)=0,g(2)=0,g(3)=0,g(4)=0,g(5)=2$

$\therefore g(x)=A(x-1)(x-2)(x-3)(x-4)$ for some constant A according to Remainder factor theorem .

$g(5)=A(4)(3)(2)(1)=2$

$\Rightarrow A=\frac { 1 }{ 12 }$

$\therefore g(x)=\frac { (x-1)(x-2)(x-3)(x-4) }{ 12 }$

$\therefore f(x)=\frac { (x-1)(x-2)(x-3)(x-4) }{ 12 } +{ x }^{ 3 }$

$\Rightarrow f(6)=226$

Oh dear. My phone wasn't showing the whole question, I kept seeing f(5) = 1. The 27 part wasn't visible. I was wondering why my answer wasn't working!

Vishnu Bhagyanath - 6 years ago

Sometimes it happens.:(

shivamani patil - 6 years ago

I solved it making a 4x4 equations system and solving for the A,B,C,D :) It´s a primitive method, but it works :)

Giovanni Dominguez - 5 years, 8 months ago

Hi sir, would the method of differences be a more of less efficient way of solving (in your opinion)?

Yee-Lynn Lee - 5 years, 11 months ago

First of all if you are referring me then don't call me sir title just I am not still eligible to called be as sir:).You can use method of differences but this seems more efficient than method of differences. Sometimes we may misplace values in that table .so in my opinion answer is yes to your question.

shivamani patil - 5 years, 11 months ago

Oh, sorry! >__< But, thanks anyway! :)

Yee-Lynn Lee - 5 years, 11 months ago

@Yee-Lynn Lee – Nevermind welcome.xD

shivamani patil - 5 years, 11 months ago

Devendra Kumar Singh
Sep 17, 2015

Same method as that of Shivamani Patil.

Poly functional

The answer is 226.

2 solutions

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