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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