Question

Show that ab is a factor of

(a+b)^n - (a^n + b^n ) ​

Answer 1

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$<marquee>DynamiteDoll</marquee>$

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Given expression

$\bf\pink{\left(a + b\right)}^{n}{-}{\left({a}^{n}{+}{b}^{n}\right)}$

let $\mathcal\pink{n}$ be an even number

Take $\bf\pink{n}\green{=}\pink{2}$

$\mathbb\blue{=}$ $\bf\pink{\left(a + b\right)}^\pink{2}\pink{-}\pink{\left({a}^{2}\pink{+}{b}^{2}\right)}$

$\mathbb\blue{=}$ $\bf\pink{a}^{2}\pink{+}\pink{2ab}\pink{+}\pink{b}^{2}\pink{-}\pink{a}^{2}\pink{-}\pinl{b}^{2}$

$\mathbb\blue{=}$ $\bf\pink{2ab}$

Hence ,

ab is a factor of the given expression because it is exactly divisible by ab

when ' n ' is an even number

Let n be an odd number

Take $\bf\pink{n}\green{=}\pink{3}$

$\mathbb\blue{=}$ ( a + b )³ - ( a³ + b³ )

$\mathbb\blue{=}$ a³ + b³ + 3ab ( a + b ) - a³ - b³

$\mathbb\blue{=}$ $\bf\pink{3ab\left(a + b\right)}$

Hence ,

ab is a factor of the given expression because it is exactly divisible by ab

When ' n ' is an odd number .

:. For all values of n , ab is a factor of

$\huge\mathcal\orange{\left(a + b\right)}^{n}{-}{\left({a}^{n}{+}{b}^{n}\right)}$

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$<marquee>Thanks...⊙ิз⊙ิ♡ ▆▎ »» ♥</marquee>$

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