Question

1) If the polynomial az³+4z²+3z-4 and z³-4z+a leave the same remainder when divided by z-3 m. Find the value of a?

Answer 1

Let f(z) = az³+4z²+3z-4
zero of the polynomial (z-3) = 3
(since,z-3=0⇒z=3)
So,replacing z by 3,
f(3) = a(3)³+4(3)²+3(3)-4
⇒f(3) = 27a+36+9-4
⇒f(3) = 27a+41
Let g(z) = z³-4z+a
zero of the polynomial (z-3) = 3
(since,z-3=0⇒z=3)
So,replacing z by 3,
g(3) = (3)³-4(3)+a
⇒g(3) = 27-12+a
⇒g(3) = 15+a
Given that the two polynomials leaves same remainder when divided by z-3
So,g(3)=f(3)
⇒15+a = 27a+41
⇒15-41 = 27a - a
⇒-26 = 26a
⇒a = -1.