<p>I need help with a specific type of question. No matter what the standardized math test (PSAT, SAT II, ACT) I always get stumped by a question similar to this one! I don't really understand the connection between the question and answers and how to solve. Thanks!</p>
<p>Which of the following is true for all consecutive integers m and n such that m<n?</p>
<p>F. m is odd
G. n is odd
H. n-m is even
J. n^2-m^2 is odd
K. m^2+n^2 is even</p>
<p>and if you want to be even more awesome:</p>
<p>For all x in the domain of the function (x+1)/(x^3-x), this function is equivalent to:</p>
<p>F. (1/x^2)-(1/x^3)
G. (1/x^3)-(1/x)
H. (1/(x^2)-1) <that is 1 divided by: x squared -1 ...if i wrote it wrong
J. (1/(x^2)-x)
K. (1/x^3)</p>
<p>are these the same type of problem? or does the second one have to do with synthetic division?</p>
<p>The answer to the first problem is J. The easiest way to do this problem is to assign the variables ‘m’ and ‘n’ a number. The problem asks for “consecutive integers,” or two numbers that are in order (1,2 or 56,57 for example.)</p>
<p>‘m’ must be less than ‘n’ so let’s assign m=1 and n=2</p>
<p>Now try the answers:
F. In this case m is odd, but what if we choose m=2 and n=3? It wouldn’t be odd.
G. n is not always going to be odd either.
H. 2-1=1 which is not odd
J. 2^2 - 1^3 = 4-1 = 3, so this is the answer.
K. 1^2 + 2^2 = 1+4 = 5, which is not even.</p>
<p>The answer to the second problem is also J. Most of the time they give this kind of equation where a simple “x+1” is on the top and a more complicated “x^3-x” is on the bottom they want you to simplify the bottom until you get a “x+1” that you can cancel out the top of the equation with. </p>
<p>The first thing you do is simplify the bottom:
x^3 - x
You can take an x out, leaving:
x(x^2 - 1)
“x^2 - 1” is a perfect square, factoring it makes its “(x-1)(x+1)” so:
x(x-1)(x+1) is on the bottom.
The “x+1” on the top of the equation can cancel the “x+1” on the bottom of the equation:
1/x(x-1) remains
Multiply the bottom out again and you are left with your answer:
1/x^2-x</p>