Question

If the polynomials az³ + 4z² + 3z – 4 and z³ – 4z + a leave the same remainder when divided by z – 3, find the value of a.

Answer 1

Zero of the polynomial,

g₁(z) = 0

z-3 = 0

z = 3

Therefore, zero of g(z) = – 2a

Let p(z) = az³ + 4z² + 3z - 4

So, substituting the value of z = 3 in p(z), we get,

p(3) = a(3)³ + 4(3)² + 3(3) - 4

⇒p(3) = 27a+36+9-4

⇒p(3) = 27a+41

Let h(z) = z³ - 4z + a

So, substituting the value of z = 3 in h(z), we get,

h(3) = (3)³ - 4(3) + a

⇒h(3) = 27-12+a

⇒h(3) = 15+a

According to the question,

We know that,

The two polynomials, p(z) and h(z), leaves same remainder when divided by z-3

So, h(3)=p(3)

⇒15+a = 27a+41

⇒15-41 = 27a – a

⇒-26 = 26a

⇒a = -1