Practice for Swapnil!
If a polynomial ,when divided by ,leaves a remainder of ,and when divided by ,leaves a remainder of , then what is the remainder when is divided by ,if the answer is of the form, , then what is the value of ?
The answer is 1.
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1 solution
We understand that when a polynomial is divided by a linear factor(like (x-1) or (x-2)) the we get a constant (independent of x). But what happens when we divide it by a quadratic factor, say (x-1)(x-2) or x^2+x+1. Then, we get the remainder a linear function (of the form mx+c). In general if we divide a polynomial(divident) with another polynomial(divisor) of degree n, then the remainder polynomial will have a degree of n-1. Thus we may assume, P(x)=(x-a)(x-c)Q(x)+mx+n. Thus ma+n=b,mc+n=z. On solving these simultaneous linear equations involving variables m and n, we get remainder function as, (b(x-c)+z(a-x))/(a-c). I think the maker of the question meant "px" instead of "p".