The problem with which I’m having some difficulty, taken from the chapter on the Passport to Advanced Math section in The Official SAT Study Guide book (page 229), is as follows:

(y^5) – (2y^4) – (cxy) + (6x)

In the polynomial above, “c” is a constant. If the polynomial is divisible by “y – 2,” what is the value of “c”?

Specifically, how does one factor “y – 2” from “- (cxy) + (6x)”?

Thank you for all your help, Mike.

First, an apology: while I aim to answer Q&A questions within 24 hours, I’m late on this one. Sorry!

This is a tricky one, and pretty unlike anything I’ve seen on a real test. For that reason, I wouldn’t worry too much about it. As you note, factoring y-2 out of the first part is easy enough.

    \begin{align*}y^5-2y^4-cxy+6x\\y^4(y-2)-cxy+6x\end{align*}

From there, you have to get creative. The way I think through it is that I know y-2 must be a factor of -cxy+6x. That means I know (y-2)(\text{SOMETHING})=-cxy+6x.

Figuring out the SOMETHING requires recognizing a few things:

  • It must contain x (both terms on the right contain x).
  • It must contain -3 (the second term gets multiplied by -2 and ends up positive 6x).

So what if we say that SOMETHING equals -3x ?

    \begin{align*}(y-2)(-3x)&=-cxy+6x\\-3xy+6x&=-cxy+6x\\-3xy&=-cxy\\3&=c\end{align*}

Does that help?

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