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!
Answers
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 !!!
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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