# The Polynomials Ax3 + 3x2 − 3 and 2x3 − 5x + A When Divided by (X − 4) Leave the Remainders R1 And R2 Respectively. Find the Values of the Following Cases, If 2r1 − R2 = 0. - Mathematics

The polynomials ax3 + 3x2 − 3 and 2x3 − 5x + a when divided by (x − 4) leave the remainders R1 and R2 respectively. Find the values of the following cases, if 2R1 − R2 = 0.

#### Solution

Let us denote the given polynomials as

f(x) = ax^3 + 3x^2 -3

g(x) = 2x^3 - 5x + a,

 h(x) = x-4

Now, we will find the remainders R1and R2 when f(x) and g(x)respectively are divided by h(x).

By the remainder theorem, when f(x)is divided by h(x) the remainder is

R_1 = f(4)

 = a(4)^3 + 3(4)^2 -3

 = 64a + 48 - 3

 = 64a + 48

By the remainder theorem, when g(x) is divided by h(x) the remainder is

R_2 = g(4)

2(4)^3 - 5(4) + a

128 - 20

 a+108

By the given condition,

2R1 − R2 = 0

⇒ 2(64a + 45) - (a+ 108) = 0

⇒              128a + 90 - a - 108 = 0

⇒                                                                      127a = 18

 ⇒                                                                                 a = 18/127

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#### APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 6 Factorisation of Polynomials
Exercise 6.3 | Q 12.3 | Page 15