What Am I Missing?

This math team problem is giving me a headache. It is one of 20 problems from the 1999 ICTM State Competition for Algebra II. The students have 50 minutes. I’ve spent hours on this one.

Given, f\left( x \right)=x^{1998}, g\left( x \right)=x^{2}-5x+6, Let R\left( x \right) be the remainder of \frac{f\left( x \right)}{g\left( x \right)}, the y-intercept of which can be expressed as ab\left( a^{c}-b^{c} \right). Find b-a+c.

I think there must be part of the remainder theorem I’ve either forgotten or never learned. I’ve backed into the solution1, I just don’t know why I can do it.


1The answer is in comments. I don’t want to ruin anyone’s fun.

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4 Responses to What Am I Missing?

  1. Jackie says:

    The answer is 1997, with a = 2, b = 3, and c = 1998.

  2. Factor g(x)=(x-2)(x-3). Then we have

    f(x) = x^{1998} = Q(x)(x-2)(x-3) + (mx+n)

    because the remainder must be a linear polynomial. Comparing constant terms, we must have

    n = -(-2)(-3)k

    as the y-intercept you’re interested in. Any clearer yet?

  3. Jackie says:

    That’s how I backed into the solution by testing f\left( x \right)=x^{4}, x^{5}

    Thank you.

  4. Pingback: Not Knowing « Continuities

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