The remainder of a polynomial f(x) in x are 10 and 15 respectively when f(x) is divided by (x-3) and (x-4).Find the remainder when f(x) is divided by (x-3)(-4)?

The remainder of a polynomial f(x) in x are 10 and 15 respectively when f(x) is divided by (x-3) and (x-4).Find the remainder when f(x) is divided by (x-3)(x-4)?

1 Answer
Feb 2, 2018

5x-5=5(x-1).

Explanation:

Recall that the degree of the remainder poly. is always

less than that of the divisor poly.

Therefore, when f(x) is divided by a quadratic poly.

(x-4)(x-3), the remainder poly. must be linear, say, (ax+b).

If q(x) is the quotient poly. in the above division, then, we

have, f(x)=(x-4)(x-3)q(x)+(ax+b)............<1>.

f(x), when divided by (x-3) leaves the remainder 10,

rArr f(3)=10....................[because," the Remainder Theorem]".

Then, by <1>, 10=3a+b....................................<2>.

Similarly,

f(4)=15, and <1> rArr 4a+b=15....................<3>.

Solving <2> and <3>, a=5, b=-5.

These give us, 5x-5=5(x-1) as the desired remainder!