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Grade 10Algebra

f(x)/x-3 gives remainder 15
f(x)/(x-1)² gives remainder 2x+1
then find remainder when f(x) is devided by (x-3)(x-1)²

satyarth shankar

8 Years agoGrade 10

1 Answer

Md Shaquib Ansari

8 Years ago

$Given,\\ f(x)=(x-3)P(x)+15 \\ f(3)=15\\ f(x)=(x-1)^{2}Q(x)+2x+1\\ f(1)=3\\ f`(x)=(x-1)^{2}Q`(x)+2(x-1)Q(x)+2\\ f`(1)=2\\ \\ Suppose, \\ f(x)=(x-3)(x-1)^{2}R(x)+Ax^{2}+Bx+C\\ Then,\\ f(1)=A+B+C=3\\ f(3)=9A+3B+C=15\\ f`(1)=2A+B=2\\ Solving,\\ A=2,B=-2,C=3\\ Remainder\\ 2x^{2}-2x+3$

Simple polynomial property is used here.

First calculated f(1),f(3) and since (x-1)² is involved this is the hint that f’(x) is going to be involved.

So calculated f’(1).

Now assume when f(x) is divided by (x-1)²(x-3) remainder is Ax²+Bx+C and using the previous three conditions got the values of A,B,C.

That’s it.

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