the polynomial x cube + 4 x square - 11 ax - 26 and 6 x cube + 17 X square + X - 8 are divided by X + 2 then remainder are same find the value of a
Answers
Answer:
Required value of a is 14 / 11.
Step-by-step explanation:
It is given that the two different polynomials( x^3 + 4x^2 - 11ax - 26 and 6x^3 + 17x^2 + x - 8 ) leave the same remainder, if divided by ( x + 2 ).
Let the remainder be r, when x^3 + 4x^2 - 11ax - 26 is divided by ( x + 2 ).
According to this question, both the polynomials are leaving the same reminder, so the remainder should be r, if 6x^3 + 17x^2 + x - 8 is divided by ( x + 2 ), since x^3 + 4x^2 - 11ax - 26 is leaving r as remainder when divided by ( x + 2 ).
By Remainder theorem : -
( x + 2 ) = { x - ( - 2 ) } . Thus, substituting the value of x as - 2 in the polynomials for the value of r.
Substituting in x^3 + 4x^2 - 11ax - 26 :
= > ( - 2 )^3 + 4( - 2 )^2 - 11a( - 2 ) - 26 = r
= > - 8 + 4( 4 ) + 22a - 26 = r
= > - 8 + 16 + 22a - 26 = r
= > 8 - 26 + 22a = r
= > - 18 + 22a = r ...( i )
Substituting in 6x^3 + 17x^2 + x - 8
= > 6( - 2 )^3 + 17( - 2 )^2 + ( - 2 ) - 8 = r
= > 6( - 8 ) + 17( 4 ) - 2 - 8 = r
= > - 48 + 68 - 10 = r
= > 10 = r ...( ii )
Comparing the values of r from ( i ) and ( ii ) : -
= > 10 = r = - 18 + 22a
= > 10 = - 18 + 22a
= > 10 + 18 = 22a
= > 28 = 22a
= > 28 / 22 = a
= > 14 / 11 = a
Hence the required value of a is 14 / 11.