Question

If x + 1 is a factor of 2x³ + ax² + 2bx + 1, then find the values of a and b given that 2a - 3b = 4

Full Explanation needed!

Answer 1

Hi dear !!!

x + 1 is factor so it satisfy the equation

2 x³ + ax² + 2bx +1 = 0

f(-1) = > 2(-1)³ + a (-1)² + 2× b -1 +1 = 0


-2 + a -2b +1 = 0


a -2b = 1. --------------------(1)


and

2a - 3b = 4 -------------------(2)


from (1 ) and (2) we have


2a - 4b = 2 { multiply ( 1) by 2 }

2a -3b = 4

b = 2

and


on putting the value of b in equation (1)

we get

a -2b = 1
a- 4 = 1

a = 5


hence a = 5 and b = 2


hope it help you dear !!!

thanks !!!

Answer 2

Let us assume f ( x ) a function such that :

f ( x ) = 2 x³ + a x² + 2 b x + 1

x + 1 is a factor of f ( x ).

By Factor Theorem :     f ( - 1 ) = 0

                                   ==> 2 x³ + a x² + 2 b x + 1 = 0 [ when x = - 1 ]

                                   ==> 2 ( - 1 )³ + a ( - 1 )² + 2 b ( - 1 ) + 1 = 0

                                   ==> - 2 + a - 2 b + 1 = 0

                                   ==> a - 2 b - 1 = 0

                                   ==> a - 2 b = `1 ............................(1)

Multiplying (1) by 2 we get :

                                   ==> 2 a - 4 b = 2.........................(2)

Given :

           2 a -  3 b = 4 ................................(3)

Subtracting (3) from (2) we get :

                                                 2 a - 4 b - 2 a + 3 b = 2 - 4

                                       ===>  - b = - 2

                                       ===>   b = 2

From (1) we know that :

                                    a - 2 b = 1

                                 ==> a - 2 × 2 = 1 [ b = 2 ]

                                 ==> a - 4 = 1

                                 ==> a = 5

The values are as follows :

a = 5

b = 2

Hope it helps ya :-)

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