Question

If
x 3 + \frac{1}{x {3} } = 18
then prove that
x + \frac{1}{x} = 3

Answer 1

Answer:

Step-by-step explanation:

Given,

     x^3 + 1 / x^3 = 18

Let x + 1 / x = a

Cube on both sides of x + 1 / x = a,

= > ( x + 1 / x )^3 = a^3

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From the properties of expansion, we know : -

( a + b )^3 = a^3 + b^3 + 3ab( a + b )

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= > x^3 + 1 / x^3 + 3( x + 1 / x )( x × 1 / x ) = a^3

= > 18 + 3( a )( 1 ) = a^3

= > 18 + 3a = a^3

= > 0 = a^3 - 3a - 18

Checking whether ( a - 3 ) is a factor of this polynomial or not, checking :

= > p( 3 ) = ( 3 )^3 - 3( 3 ) - 18

= > p( 3 ) = 27 - 9 - 18

= > p( 3 ) = 0

Hence, ( a - 3 ) is a factor. So

= > a = 3

= > x + 1 / x = 3

Proved.