Question

$if \: x + 2k \: is \: a \: factor \: of \: {x}^{5} - 4 {k}^{2} {x}^{3} + 2x + 2k + 3.Find the value of k$

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Answer 1

Here , in the above Question , the following information is given -

( x + 2k ) is a factor of x⁵ - 4k²x³ + 2x + 2k + 3.

To find -

Find the value of k .

Solution -

Here , the given equation , f ( x ) = x⁵ - 4k²x³ + 2x + 2k + 3 .

x is the variable whereas k is the constant .

Now , g ( x ) = x + 2k .

Here , g ( x ) is a factor of f ( x ) .

So ,

when x = -2 k , f ( -2k ) = 0

=> f ( - 2k ) = ( - 2k )⁵ - 4 ( k )² ( - 2k )³ + 2 ( -2 k ) + 2k + 3

=> f ( - 2k ) = -32 k⁵ + 32 k⁵ - 4k + 2k + 3

=> f ( - 2k ) = -2 k + 3.

But ,

f ( - 2k ) = 0

=> - 2k + 3 = 0

=> - 2k = - 3 .

=> k = 3/2 .

Thus , the required value of k is 3/2 .

This is the answer .

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