Question

When the polynomial $f(x)$ is divided by $x - 3,$ the remainder is 15. When $f(x)$ is divided by $(x - 1)^2,$ the remainder is $2x 1.$ Find the remainder when $f(x)$ is divided by $(x - 3)(x - 1)^2.$

Answer 1

The remainder when f(x) is divided by (x - 3)(x - 1)^2 is (6x - 3).

What is the remainder theorem for polynomials?

If there is a polynomial p(x), and a constant number 'a', then

$\dfrac{p(x)}{(x-a)} = g(x) + p(a)$

where g(x) is a factor of p(x)

Here, f(x) is the given polynomial.

By Remainder Theorem,

When divided by (x-3),

f(3) = 15........(1)

When divided by (x-1)²,

f(1) = 2x - 1........(2)

Another polynomial is (x - 3)(x - 1)²

= (x - 3)(x - 1)(x -1)

So,

f(x) = (x - 3)(x - 1)(x -1)Qx + (ax+b)

For f(1),

2x + 1 = a + b

3 = a + b

For f(3),

15 = 3a + b

or, 15 = 3a + 3 - a

or, 15 = 3 + 2a

or a = 6

Also, b = -3

Hence, the remainder is (6x - 3).

Learn more about the remainder :

brainly.com/question/15091928

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