Polynomial

Algebra Level 3

Consider a monic polynomial $f(x)$ of degree 4.

If $f(1) = 1, f(2) = 2, f(3) = 3,$ and $f(4) = 4$ , then what is the value of $f(6)$ ?

The answer is 126.

3 solutions

Abdur Rehman Zahid
Mar 9, 2016

We can use the Remainder factor theorem to solve this problem.

Let $g(x)=f(x)-x$ .Then $g(x)=0$ for $x=1,2,3,4$ .Hence $g(x)=(x-1)(x-2)(x-3)(x-4)$ (Note that $f(x)$ is monic, so $g(x)$ is monic as well).Hence: $f(x)=(x-1)(x-2)(x-3)(x-4)+x$

Simply plugging in $x=6$ gives $f(x)=\boxed{126}$

Moderator note:

Great solution. RFT provides a quick way of understanding the problem.

Kshitij Goel
Sep 24, 2015

Polynomial is greater than Of 2 degree So polynomial can't be (x)

So the polynomial satisfying all relations is

(x-1)(x-2)(x-3)(x-4)(x-5)+x

So by putting x=6 This equals 126

Plz upvote guys if u liked it

Kshitij Goel - 5 years, 8 months ago

Not liked only. I liked very much.

Priyanshu Mishra - 5 years, 7 months ago

I took $f(x)=2(x-1)(x-2)(x-3)(x-4)+x$ , with $f(6)=246$ ... I think that works too ;)

Otto Bretscher - 5 years, 8 months ago

Ya we can take various polynomials like this ..but this is the simplest one which I took

Kshitij Goel - 5 years, 8 months ago

Sure... but the problem makes no sense as posed

Otto Bretscher - 5 years, 8 months ago

Yeah You are right, the problem must mention the leading coefficient.

Kushagra Sahni - 5 years, 8 months ago

Sanyam Goel
Apr 8, 2016

I think the problem is overrated ..... :(

Polynomial

The answer is 126.

3 solutions

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