Q. 104.0( 10 Votes )

# If the polynomial

Answer :

Let, p (x) = *ax*^{3}+3*x*^{2}-3 and q (x) = 2*x*^{3}-5*x*+*a* be the given polynomials.

Now,

R_{1} = Remainder when p (x) is divided by (x – 4)

= p (4)

= a (4)^{3} + 3 (4)^{2} – 3 [Therefore, p (x) = *ax*^{3}+3*x*^{2}-3]

= 64a + 48 – 3

R_{1} = 64a + 45

And,

R_{2} = Remainder when q (x) is divided by (x – 4)

= q (4)

= 2 (4)^{3} – 5 (4) + a [Therefore, q (x) = 2*x*^{3}-5*x*+*a*]

= 128 – 20 + a

R_{2} = 108 + a

(i) Given condition is,

R_{1} = R_{2}

64a + 45 = 108 + a

63a – 63 = 0

63a = 63

a = 1

(ii) Given condition is R_{1} + R_{2} = 0

64a + 45 + 108 + a = 0

65a + 153 = 0

65a = -153

a =

(iii) Given condition is 2R_{1} – R_{2} = 0

2 (64a + 45) – (108 + a) = 0

128a + 90 – 108 – a

127a – 18 = 0

127a = 18

a =

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