Its not f(x)=x

Algebra Level 4

f(x) is a polynomial of degree 5 with leading coefficient 1 (coefficient of x^5). f(1)=1, f(2)=2, f(3)=3, f(4)=4, f(5)=5, Then find the value of f(6)

The answer is 126.

3 solutions

Brian Charlesworth
Nov 29, 2014

Let $g(x) = f(x) - x$ . Then $g(x) = 0$ for $x = 1,2,3,4,5$ .

Thus $g(x)$ is a polynomial of degree $5$ with a leading coefficient of $1$ with roots $x = 1,2,3,4,5$ , and thus, by the Fundamental Theorem of Algebra, can be written

$g(x) = (x - 1)(x - 2)(x - 3)(x - 4)(x - 5)$ .

Plugging in $x = 6$ gives us that $g(6) = 5*4*3*2*1 = 120$ , and so

$f(6) = g(6) + 6 = \boxed{126}$ .

I want ask why i cant use this method by finding all constant of this x^5+ax^4+bx^3+cx^2+dx+e and then substitute 6 into the equation ?because the answer that it appears is 750 instead of 126

Loki Come - 6 years, 6 months ago

did the exact same! Well, except the fact that

$g(x) = a(x-1)(x-2)(x-3)(x-4)(x-5)$

and then as it is monic, a can only be equal to 1.

Kartik Sharma - 6 years, 6 months ago

Divyansh Chaturvedi
Nov 29, 2014

Let the polynomial f(x) be equal to

(x-1)(x-2)(x-3)(x-4)(x-5) + x. Note that it satisfies all the given conditions i.e

f(1) = 1 f(2) = 2 f(3) = 3 f(4) = 4 And f(5) = 5

So putting the value of x as 6 in f(x) we get

5* 4* 3* 2* 1 + 6 = 126 So 126 is the answer...

Why should the polynomial equal the above given expression?

Krishna Ar - 6 years, 6 months ago

Afreen Sheikh
Jan 5, 2015

as given f(x)=x hence 1,2,3,4,5 are roots of f(x)-x=0 and equation can be written as f(x)-x=(x-1)(x-2)(X-3)(x-4)(x-5) thus for x=6 f(6)=6+(6-1)(6-2)(6-3)(6-4)(6-5) f(6)=126

Its not f(x)=x

The answer is 126.

3 solutions

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