11/7/2023 0 Comments Sat math practice test 2016Since the SAT has to give us at least one valid answer, (D) is correct by the process of elimination. Since p(5) = 0 doesn't have to be true, choice (A) is wrong.Ĭhoices (B) and (C) are wrong for the same reason. We do know, however, that the problem asks us what must be true. The problem doesn't tell us what p(5) is, so we don't know if it's zero. That means that if x − 5 is really a factor of p(x), then p(5) = 0. P(x) = ( x − 5)( x blah blah blah)( x blah blah blah)įrom the factored form, we can see that when x = 5, the polynomial has to equal zero, since ( x − 5) = 0 when x = 5. In other words, if we factor p(x), we should get something like this: Method #3: Test the answer choices by factoring.Ĭhoice (A) asks whether x − 5 is a factor of p(x). While that's true, it would have made the problem trickier and would also have failed to eliminate choice (A), making it necessary to test yet another polynomial to decide whether (A) or (D) is the correct answer. Since the test has to give us a valid answer choice, (D) has to be the correct answer.Īn advanced student will remind me that I could have picked other polynomials that satisfy p(3) = -2, such as p(x) = x -5. Remember, the question is asking what must be true for p(x), so showing that (D) is true for p(x) = -2 doesn't prove that it's true for other possible polynomials where p(3) = -2.Įven though we're not 100% sure that (D) is true, we've already eliminated choices (A), (B), and (C). This is exactly what choice (D) tells us should happen. X - 3 goes into -2 zero times with -2 left over, which is another way of saying that -2 is the remainder. Let's try dividing our very simple p(x) by x - 5. For example, 25/5 has no remainder because 5 is a factor of 25, but 29/5 has a remainder of 4 because 5 isn't a factor of 29. When you divide a number by its factor, you get a remainder of zero. The question uses the word must, so it doesn't matter how simple our polynomial is as long as p(3) = -2.Ĭhoice (A) asks whether x - 5 is a factor of p(x). This may look like cheating, but notice that when x = 3, p(x) = -2, so it meets the requirements of the problem. I'm going to create the simplest p(x) I can think of that makes p(3) = -2: The question does tell us that p(3) = -2, but it doesn't give us any other requirements. Therefore, if Question 29 is asking what must be true about p(x), we can just make up a polynomial and test the answer choices! If the sentence is true, it has to apply to every swan in existence. If something must be true, it has to be true for each and every example.Ī single black swan would disprove the sentence. Just how do you test answer choices when the problem doesn't even give you a polynomial to work with? Method #2: Test the answer choices by choosing a polynomial. While I understand the value of proving mathematical theorems, normal people like you and I shouldn't have to improvise proofs on a timed test of basic math concepts. She proceeds to re-create the proof for the Polynomial Remainder Theorem from scratch.įinished with her test, Miss Bach jets off to mail in her Mensa application. Esther walks into the test not knowing the textbook method for solving the problem. It works pretty well for Esther Godel Bach, that math genius who doesn't prep for the SAT. The College Board's explanation (page 41) relies on Method #1. Method #1: Prove a mathematical theorem while taking your test. You can tackle this intimidating-looking problem four different ways, only one of which is as complicated as the problem seems. Which of the following must be true about p(x)?ĭ) The remainder when p(x) is divided by x − 3 is −2. Solution for New SAT Practice Test #1, Calculator-Based Math, Problem 29 (page 53)įor a polynomial p(x), the value of p(3) is −2.
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