Let P(x) be a polynomial of n degree , prove P(a)-p(b) is divisible by a-b

Asked Aug 08 '21 at 06:01

Active Aug 08 '21 at 08:39

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Let p(x) be a polynomial of n degree prove that p(a)-p(b) is divisible by a-b My first thought was to substitute a and b in the place of x and form a general expression then try to prove it but I didnt get any answer afterward Could anyone help me out

edited Aug 08 '21 at 08:39

asked Aug 08 '21 at 06:01

Aayat khan

3

Are $P$ and $p$ the same polynomial? – ndhanson3 Aug 08 '21 at 06:03
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Hint: try to prove $a-b|a^n-b^n$. – Kenta S Aug 08 '21 at 06:03
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Alternative approach: show that $x-b\mid P(x)-P(b)$ as polynomials (also I assume that the polynomials should be in $\Bbb Z[x]$) – leoli1 Aug 08 '21 at 06:06
Welcome to MSE. Please use MathJax to format math on this site. To begin with, enclose all math expressions (including numbers) in $ signs. For example, $x_1^2$ will give you $x_1^2$. You'll get a much better response if your posts are easy to read. – saulspatz Aug 08 '21 at 06:18
I tried but couldn't get to it anyone ? – Aayat khan Aug 08 '21 at 06:21
@Aayatkhan induction. – Kenta S Aug 08 '21 at 08:41

Let P(x) be a polynomial of n degree , prove P(a)-p(b) is divisible by a-b

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