Solving polynomial

Algebra Level 4

Consider the polynomial

$f(x)= x^4 + ax^3 + bx^2 + cx + d.$

Given that $f(k) = k$ for $k= 1,2,3,4,$ then find $f(5)$ .

The answer is 29.

1 solution

Brian Charlesworth
Oct 28, 2016

Let $g(k) = f(k) - k$ . Then $g(k)$ is a 4th degree polynomial with roots $1,2,3$ and $4$ , and so

$g(k) = (k - 1)(k - 2)(k - 3)(k - 4)$ . Plugging in $k = 5$ gives us that

$g(5) = 4 \times 3 \times 2 \times 1 = 24$ , and so $f(5) = g(5) + 5 = \boxed{29}$ .

I think the solution is very good. I am also extremely impressed about the fact you are 51,not to be rude but the fact that at such a old age as the oldest I have seen was in the 30's,you being 51 was shocking. I know this isn't a usual comment but it is amazing that someone so old has such determination to make solutions. I am not praising you, I truly consider it a great achievement.

Razzi Masroor - 4 years, 7 months ago

Haha Thanks! Well, age is a state of mind as much anything, in which I feel like I'm still in my 30's. :) I'm seeing more and more older participants on Brilliant, which is a good thing. The oldest I've seen is 88, and he posts some great solutions!

Brian Charlesworth - 4 years, 7 months ago

I was amazed about you being 51 and now you tell me you have seen an 88 year old!

Razzi Masroor - 4 years, 7 months ago

Solving polynomial

The answer is 29.

1 solution

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