PWN p. 151 #8 question

In PWN the SAT Math Guide (4th Ed, first printing p. 151), Polynomials chapter question #8, you explain the answer by graphing on calculator. Could you explain the answer to this question without graphing, in particular, why A, C and D are true and why B is false? Many thanks!

Sure! First, recognize that *g*(*x*) is factorable:

From there, you can see that *g*(*x*) has two real zeros (2 and –5) so you can eliminate C.

Then, recognize that *g*(*x*) is a factor of *f*(*x*) as choice A says:

Obviously that confirms that choice A is true, so you can eliminate A. However, that *also* confirms that choice B is false! Remember how we already factored *g*(*x*)!

*f*(*x*) is divisible by (*x* + 5), but definitely *not* by (*x* – 5). You know from the way the question is constructed that only one choice will be false, so once you’ve found it, you can stop working and start bubbling. 🙂

Since you asked, though, we can also show that D is true from what we’ve already figured out using substitution. We already said , so we can substitute thusly:

We can factor a *g*(*x*) out of each of those terms!

Therefore, choice D is also true.