Question

For what value of a is 2x³+2ax²+11x+a+3 exactly divisible by (2x-1)

Answer 1

Hi,

Let p( x ) = 2x³ + 2ax² + 11x + a + 3

It is given that ,

p( x ) is exactly divisible by ( 2x - 1 ).

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By factor theorem :


If p( x ) is exactly divisible by ( x - a )

then the remainder is p ( a ) = 0

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Remainder = 0

p( 1/2 ) = 0

2( 1/2)³ + 2a( 1/2 )² + 11( 1/2 ) + a + 3 = 0

2/8 + 2a/4 + 11/2 + a + 3 = 0

1/4 + a/2 + 11/2 + a + 3 = 0

LCM = 4

( 1 + 2a + 22 + 4a + 12 )/4 = 0

6a + 35 = 0

6a = -35

a = -35/6

I hope this helps you.

: )

Answer 2

Hiii friend,

(2X-1) is a factor of the given polynomial.

(2X-1) => 0

X = 1/2

Putting X = 1/2 in P(X).

P(X) => 2X³+2AX²+11X+A+3

P(1/2) => 2 × (1/2)³ + 2 × A × (1/2)² × 11 × 1/2 + A +3...

=> 2 × 1/8 + 2 × A × 1/4 + 11/2 +4A+ 3

=> 1/4 + A/2+ 11/2 +4A+3

=> 1/4 + A/2 + 11/2 +4A+ 3

=> (1 × 1 + A × 2 + 11 × 2 + A + 3×4)/4

=> (1+2A+22+4A+12)/4 = 0

=> 6A + 35 = 0

=> A = -35/6


HOPE IT WILL HELP YOU..... :-)