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)?

Question

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)?

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Answer 1 · 2018-02-02T17:44:43

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

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!