Q. 54.6( 8 Votes )

For what value of a, the divisions of two polynomials (ax3 + 3x2 – 3) and (2x3 – 5x + a) by (x – 4) give the same remainder —let us calculate and write it.

Answer :

Let the two polynomials be,


P(x) = ax3 + 3x2 – 3 …(i)


Q(x) = 2x3 – 5x + a …(ii)


Now, we understand by the question that,


P(x) and Q(x) divided by (x – 4) gives the same remainder.


We need to find zero of the linear polynomial, (x – 4).


To find zero, put (x – 4) = 0


x – 4 = 0


x = 4


By Remainder theorem that says, f(x) is a polynomial of degree n (n ≥ 1) and ‘a’ is any real number. If f(x) is divided by (x – a), then the remainder will be f(a).


Here, a = 4.


This means, remainder when P(x) is divided by (x – 4) is P(4).


Remainder = P(4)


Remainder = a(4)3 + 3(4)2 – 3


Remainder = 64a + 48 – 3


Remainder = 64a + 45 …(iii)


And remainder when Q(x) is divided by (x – 4) is Q(4).


Remainder = Q(4)


Remainder = 2(4)3 – 5(4) + a


Remainder = 128 – 20 + a


Remainder = 108 + a …(iv)


When P(x) and Q(x) are divided (x – 4) , they leave same remainder.


Comparing equations (iii) and (iv), we have


64a + 45 = 108 + a


64a – a = 108 – 45


63a = 63



a = 1


Thus, a = 1.


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