A Question

Let p(x) be a polynomial of degree 8, such that p(k)= $\frac{1}{k}$ for k=1,2,3,4,5,6,7,8,9. What is the value of p(10)?

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

The mualphatheta link has (almost) this question set as a problem - the answer link did not work for me. Note that $xp(x)-1$ is a degree $9$ polynomial with $1,2,3,4,5,6,7,8,9$ as zeros, and hence $xp(x) - 1 \; = \; A(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)$ Putting $x=0$ we see that $-1 \,=\, -A\times9!$ and so $A = \tfrac{1}{9!}$ . Then $10p(10)-1 \,=\, A\times9! = 1$ , and hence $p(10) = \tfrac{1}{5}$ .

Mark Hennings - 7 years, 9 months ago

As a side note, you should verify that $\frac{ 1 + \frac{1}{9!} \prod_{i=1}^9 (x-i) } { x }$

is indeed a polynomial. This follows because the constant term in the numerator cancels out, so it is a multiple of $x$ .

It is possible (especially in scenarios where the exact roots are uncertain), for the function that you define to end up being a rational function, instead of a polynomial.

Calvin Lin Staff - 7 years, 9 months ago

I already did this when I evaluated $A$ , because I chose the value of $A$ to be such that the value of $1 + A\prod_{j=1}^9(x-j)$ when $x=0$ was $0$ , which means that this polynomial has no constant term and hence can be safely divided by $x$ .

Mark Hennings - 7 years, 9 months ago

This will do it : http://www.mualphatheta.org/problemcorner/MathematicalLog/Issues/0402/MathematicalLogProblemistSpring02.aspx

Vikram Waradpande - 7 years, 9 months ago

Try remainder theorem, although it will take many steps.

Siddharth Kumar - 7 years, 9 months ago

you can use newton-ramphson techhnique, newton forward or backward formula . it will be easy .

Ashok Kumar - 7 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$