find the value of a and b so that polynomial x^3 - ax^2 - 13x + b is exactly divisible by (x-1) as well (x-3)
Answers
Answer:
Required value of a is 3 and b is 15.
Step-by-step explanation:
Your questions needs a correction.
Correct Factors : ( x - 1 ) & ( x + 3 )
It is given that the polynomial x^3 - ax^2 - 13x + b is exactly divisible by ( x - 1 ) as well as ( x - 3 ), so by using remainder theorem, remainder will be 0 on taking the given factors.
Thus, by using remainder theorem, on taking ( x - 1 ) as a factor :
{ x - ( 1 ) }
= > f( 1 ) = 0
= > ( 1 )^3 - a( 1 )^2 - 13( 1 ) + b = 0
= > 1 - a( 1 ) - 13 + b = 0
= > 1 - a - 13 + b = 0
= > b - a - 12 = 0
= > b - 12 = a ...( 1 )
On taking ( x + 3 ) as a factor : { x - ( - 3 ) }
= > f( - 3 ) = 0
= > ( - 3 )^3 - a( - 3 )^2 - 13( - 3 ) + b = 0
= > - 27 - a( 9 ) + 39 + b = 0
= > - 27 - 9a + 39 + b = 0
= > b - 9a + 12 = 0
= > b + 12 = 9a
= > ( b + 12 ) / 9 = a ...( 2 )
Comparing the value of a from ( 1 ) & ( 2 ) :
= > b - 12 = ( b + 12 ) / 9
= > 9( b - 12 ) = b + 12
= > 9b - 108 = b + 12
= > 9b - b = 12 + 108
= > 8b = 120
= > b = 15
Substituting the value of b in ( 1 ) :
= > b - 12 = a
= > 15 - 12 = a
= > 3 = a
Hence the required value of a is 3 and b is 15.
a=0
b=12
hope this will be helpful
