Remainder when a polynomial is divided by a quadratic

Divided by (x – a)(x – b)

Let f(x) be a polynomial with real coefficients. If f(x) is divided by a quadratic (x – a)(x - b), the remainder must be a linear function and should be of the form Ax + B.

We would like to find the values of A and B and hence the remainder. Let q(x) be the quotient of division.

We write,

f(x) = (x – a)(x – b) q(x) + (Ax + B) ….. (1)

Now, f(a) = (a – a)(a – b) q(a) + Aa + B

= Aa + B ……..(2)

Similarly, f(b) = Ab + B ……..(3)

(2) – (3), f(a) – f(b) = A(a – b)

If a ¹ b,

Substitute (4) in (2) and simplify we get:

The remainder is:

An example

Find the remainder if f(x) = (x – 1)¹⁰⁰ + 2(x – 2)⁹⁹ + 6 is divided by (x – 1)(x – 2).

Solution

f(2) = (2 – 1)¹⁰⁰ + 2(2 – 2)⁹⁹ + 6 = 7

f(1) = (1 – 1)¹⁰⁰ + 2(1 – 2)⁹⁹ + 6 = 4

By (6), Remainder

Divided by (x – a)²

If a = b , the previous calculations seem not work well. The reason is that the denominators of the formula in (6) is equal to zero.

We therefore begin with:

Now,

Differentiate (7),

Hence f ’(a) = A ………(9)

Substitute (9) in (8) and simplify,

B = f(a) – a f ’(a) ……..(10)

The remainder is:

R = f ’(a) x + f(a) – af ’(a) ………(11)

Further note:

If we take:

we can get (11) from (6). You may have a try.