Question

if polynomials ax^3+3x^2-3 and 2x^3-5x+a when divided by x - 4 leaves the same remainder. Find the value of a

Answer 1

Dividing ax^3+3x^2-3 by x-4, we get the remainder as 45 + 64 a
Dividing 2x^3-5x+a by x-4, we get the remainder as a+108

Now, acc to question, a + 108 = 45 + 64 a

63 a = 63

a = 1

Answer 2

Answer:  The required value of a is 1.

Step-by-step explanation:  Given that the following polynomials leaves the same remainder when divided by (x - 4) :

$f(x)=ax^3+3x^2-3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\g(x)=2x^3-5x+a~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)$

We are to find the value of a.

Remainder theorem :  When (x - b) divides a polynomial p(x), then the remainder is p(b).

So, from (i) and (ii), we get

$f(4)=g(4)\\\\\Rightarrow a\times 4^3+3\times4^2-3=2\times4^3-5\times4+a\\\\\Rightarrow 64a+48-3=128-20+a\\\\\Rightarrow 64a-a=108-45\\\\\Rightarrow 63a=63\\\\\Rightarrow a=\dfrac{63}{63}\\\\\Rightarrow a=1.$

Thus, the required value of a is 1.