Maths Help - POLYNOMIALS?

Question

Hi, Below is a question of polynomials. Could you please help me get to know how the answer is obtained in a detailed way? Please make sure u explain the answer and its steps clearly so that i can understand it. Thanks a lot in advance!!

Qn: When a polynomial, P(x), is divided by x-a, it leaves a remainder of a^3 and when it is divided by x-b, it leaves a remainder of b^3. Find the remainder when P(x) is divided by (x-a)(x-b).

Note -> The answer to this question is (a^2 + ab + b^2)x - ab(a + b). However, I dont have a single idea as to how u get this answer. So please explain clearly. VERY CLEARLY.

Thanks.

Answer 1

Based on the given information, you have

P(x) = (x-a)Q(x) + a^3
and
P(x) = (x-b)R(x) + b^3

where Q(x) and R(x) are (unknown) polynomials of degree 2.

This gives you two equivalent expressions for P(a):
P(a)=a^3 and P(a)=(a-b)R(a)+b^3.

Because they must be equal, it follows that

(a-b)R(a) = a^3-b^3 = (a-b)(a^2+ab+b^2).

The last step is a well-known factorization of the difference of two cubes.

Dividing by a-b, you get R(a) = a^2+ab+b^2. Therefore, R(x) has the form

R(x) = (x-a)S(x) + a^2+ab+b^2

where S(x) is an unknown polynomial of degree 1.

Substitute this expression for R(x) into the above expression for P(x), and you get

P(x) = (x-b)[(x-a)S(x) + a^2+ab+b^2] + b^3
= (x-b)(x-a)S(x) + (x-b)(a^2+ab+b^2) + b^3
= (x-b)(x-a)S(x) + (a^2+ab+b^2)x - ab^2 - a^2b
= (x-b)(x-a)S(x) + (a^2+ab+b^2)x - ab(a+b).

Answer 2

Kool gurl, You have received several answers to this question, because you have asked this question twice. Have you understood?
I too had found this question baffling, but now I've got it very clear.
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