If the polynomial x⁴ – 2x³ + 3x² – ax + b is divided by (x – 1) and (x + 1) and the remainders are 5 and 19 respectively. But if that polynomial is divided by x + 2, then what will be the remainder —Let us calculate.

We have the polynomial x⁴ – 2x³ + 3x² – ax + b.

Let it be P(x), such that

P(x) = x⁴ – 2x³ + 3x² – ax + b …(i)

According to the question,

P(x) is divided by (x – 1) and (x + 1), and leaves the remainder 5 and 19 respectively.

We will use remainder theorem here.

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).

At first, let’s find the zero of the linear polynomial, (x – 1).

To find zero,

x – 1 = 0

⇒ x = 1

Now, let’s find the zero of the linear polynomial, (x + 1).

To find zero,

x + 1 = 0

⇒ x = -1

From Remainder theorem, we can say

When P(x) is divided by (x – 1), the remainder comes out to be 5.

We can also say that,

The remainder comes out to be P(1).

⇒ P(1) = 5

⇒ (1)⁴ – 2(1)³ + 3(1)² – a(1) + b = 5

⇒ 1 – 2 + 3 – a + b = 5

⇒ b – a + 2 = 5

⇒ b – a = 5 – 2

⇒ b – a = 3 …(ii)

And when P(x) is divided by (x + 1), the remainder comes out to be 19.

We can also say that,

The remainder comes out to be P(-1)

⇒ P(-1) = 19

⇒ (-1)⁴ – 2(-1)³ + 3(-1)² – a(-1) + b = 5

⇒ 1 + 2 + 3 + a + b = 5

⇒ a + b + 6 = 5

⇒ a + b = 5 – 6

⇒ a + b = -1 …(iii)

Solving equations (ii) and (iii), we get

(b – a) + (a + b) = 3 + (-1)

⇒ 2b = 3 – 1

⇒ 2b = 2

⇒

⇒ b = 1

Putting b = 1 in equation (ii),

b – a = 3

⇒ 1 – a = 3

⇒ a = 1 – 3

⇒ a = -2

The values of a = -2 and b = 1.

∵ The polynomial = x⁴ – 2x³ + 3x² – ax + b

∴ The polynomial = x⁴ – 2x³ + 3x² – (-2)x + 1

⇒ The polynomial = x⁴ – 2x³ + 3x² + 2x + 1

So, when polynomial x⁴ – 2x³ + 3x² + 2x + 1 is divided by (x + 2), the remainder can be calculated by using Remainder theorem.

First, we need to find zero of the linear polynomial, (x + 2).

To find zero,

Put (x + 2) = 0

⇒ x = -2

So, Required remainder = P(-2)

⇒ Required remainder = (-2)⁴ – 2(-2)³ + 3(-2)² + 2(-2) + 1

⇒ Required remainder = 16 + 16 + 12 – 4 + 1

⇒ Required remainder = 32 + 12 – 3

⇒ Required remainder = 44 – 3

⇒ Required remainder = 41

Thus, remainder is 41.