Hi. I have your New SAT Game Plan book. Have tried to follow your suggestions for dealing with functions. There was a question on a Princeton Review SAT Practice test my nephew took a few weeks ago. I have been tutoring him with your book. Maybe you can tell us how to approach this. We are lost.
f(x) = ax^3 + bx^2 + cx + d.
a,b,c and d are constants. If f(-5)=3, which of the following must be true about f(x)?
A) x – 3 is a factor of f(x)
B) The remainder when f(x) is divided by x + 5 is 3.
C) x + 2 is a factor of f(x)
D) x + 5 is a factor of f(x)
I have your older version book, so if it’s covered in the new book, let me know. I didn’t think the math changed that much in the new version of the test.
The math has in fact changed a lot! This question is asking about something called the remainder theorem. I’ll try to be brief…
Say you have a polynomial, f(x). If some number, c, is a root of that polynomial, that means two things:
1. f(c) = 0
2. When you divide f(x) by x-c, it divides evenly and you get no remainder.
On the other hand, if c is NOT a root, then when you divide f(x) by x-c you get a remainder. And the remainder theorem says that the remainder will also be what you get when you evaluate f(c).
So in this problem, since f(-5) is 3 (and not zero), we know that (x+5) does not divide f(x) evenly. It has a remainder of 3.
And yes, this is covered in the new book 🙂
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