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

Let p₁(z) = az³ + 4z² + 3z – 4 and p₂(z) = z³ – 4z + 0

When we divide p₁(z) by z – 3, then we get the remainder p,(3).

Now, p₁(3) = a(3)³ + 4(3)² + 3(3) – 4

= 27a + 36 + 9 – 4

= 27a + 41

When we divide p₂(z) by z – 3 then we get the remainder p₂(3).

Now, p₂(3) = (3)³ – 4(3) + a

= 27 – 12 + a

= 15 + a

According to the question,

Both the remainders are same.

p₁(3) = p₂(3)

27a + 41 = 15 + a

27a – a = 15 – 41

26a = –26

a = `(-26)/26`

a = –1