Question

$\text{For what value is the polynomial} \\ p(x) = 2 {x}^{3} - k {x}^{2} + 3x + 10 \\ \text{exactly divisible by} \: (x + 2) \text{?} \\ (a) \: \frac{1}{3} \\ (b) \: \frac{1}{ \sqrt{3} } \\ (c) \: 3 \\ (d) \: - 3$

Answer 1

Given factor : ( x + 2 )

It is given that p( x ) = 2x³ - kx² + 3x + 10 is exactly divisible by ( x + 2 ).

So the remainder should be 0, it means p( x ) = 0


Now, x + 2 = 0 , x = - 2


Therefore,
= > p( x ) = 2x³ - kx² + 3x + 10

= > p( - 2 ) = 2( - 2 )³ - k( - 2 )² + 3( - 2 ) + 10

= > 0 = 2( - 8 ) - k( 4 ) - 6 + 10

= > 0 = - 16 - 4k + 4

= > 0 = - 12 - 4k

= > 4k = - 12

$= > k = - \dfrac{12}{4}$

= > k = - 3



Hence the numeric value of used variable 'k' is - 3.


Thus,
Option ( d ) is the correct option.
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