Malfunction Funcion

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$f(x)$ is a degree $7$ polynomial such that, for all natural values of $x = 1 \rightarrow 8$ , $f(x) =x^{-1}$ .

The value of $f(0)$ can written as an irreducible fraction $\dfrac{a}{b}$ . Evaluate $a-b$ .

The answer is 481.

1 solution

Haroun Meghaichi
Jan 17, 2014

$p(x)= x f(x)-1$ , is a polynomial of degree $8$ has roots of : $\{1,2,\dots,8\}$ , So : $xf(x)-1=p(x) = c \prod_{k=1}^{8}(x-k).$ The constant term in $p(x)$ is $-1$ therefore the constant term in $p(x)$ is $1$ too this means that : $c=\frac{-1}{8!}$ . Now we have : $f(x)= \frac{p(x)+1}{x}.$ this means that $f(0)$ is the coefficient of $x$ in $p(x)$ : $f(0) = \frac{761}{280}.$

Multiple mistakes in your solution:

"The constant term in $p(x)$ is $-1$ therefore the constant term in $p(x)$ is $1$ too" // Big typo here, man! The first function named should read " $f(x)$ ", and also, $-1 \neq 1$ , so you can't say it is " $1$ too".

If $x=0$ , $\dfrac{\text{anything}}{0}$ is not defined.

I know $f(0)$ is the coefficient of $x$ in $p(x)$ , but how the hell did you find out its value from that?

Guilherme Dela Corte - 7 years, 4 months ago

Malfunction Funcion

The answer is 481.

1 solution

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