POLYNOMIALS BY KUSHAGRA SAHNI

Algebra Level 3

A polynomial p(x) when divided by x+2014 gives the remainder 2015 and when divided by x+2015 gives the remainder 2014. The remainder when p(x) is divided by (x+2014)(x+2015) is of the form (ax+b). Find a+b.

The answer is 4030.

1 solution

Priyanshu Mishra
Oct 16, 2015

Very simple.

Let the polynomial be $p(x)$ .

By hypothesis, we have

$p(x)$ = $(x+2014$ . $q$ + $2015$ )

$p(x)$ = $(x+2015$ . $k$ + $2014$ ), where $q$ and $k$ are quotients.

$\Rightarrow p(x) = (x+2015)(x+2014).z +(ax+b)$ for some quotient $z$

Now, as $p(-2014)=2015$ and $p(-2015)=2014$

$p(-2014)=b-2014a=2015$ and $p(-2015)=b-2015a=2014$ .

Solving both the equation yields $a=1$ and $b=4029$ forcing $a+b=\boxed{4030}$

POLYNOMIALS BY KUSHAGRA SAHNI

The answer is 4030.

1 solution

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