All posts by Cardinal Educational Consulting

SAT Math: The Remainder Theorem

The Remainder Theorem is not something that you will use many times when taking the SAT, but it has shown up on a couple of problems in the practice tests that have been released in The Official SAT Study Guide.

The Remainder Theorem says that if you divide some polynomial p(x) by the linear factor x – a, the remainder that you get is equal to p(a). For example, consider the problem below where the polynomial p(x) = x² – 2x – 11 is divided by x – 5 using either Long Division or Synthetic Division.

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SAT Math: Exponential Functions

One of the function types that you may see on the New SAT is the exponential function, which students are normally introduced to in Algebra II. These functions, which can be used to model population growth, to compute compound interest, and for other applications in science, have equations in the form:

f(x) = a·b^x

In this equation, a and b are constants and the variable, x, appears as an exponent.

The number a is the initial value (when x = 0). For a population growth problem, it would represent the initial population of whatever group you are looking at. For a compound interest problem, the a represents the amount of the deposit or the loan.

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ACT Math: A Challenging Arc Length & Sector Area Problem

One of our more popular posts involves Arc Lengths & Sector Areas for circles. With that in mind, we thought we’d give you a challenging problem (well, really two problems) involving this topic. Give it a try and then scroll down for the answer.

The circle below has its center at O and triangle AOB is equilateral with sides that are each 6 inches.

1.  What is the area, in square inches, of the shaded region?

A.  6π – 3√3
B.  9π – 3√3
C.  6π – 6√3
D.  6π – 9√3
E.  9π – 9√3

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SAT Math: Statistics and Histograms

Last week’s post looked at Statistics and Frequency Tables. Today’s let’s look at a similar problem where the data is instead presented to us in a histogram. If you didn’t read last week’s post about frequency tables, you might want to start there first.

Let’s look at three problems based on data in a histogram.

The next three problems are based on the histogram below.

The library at the local college is trying to determine how frequently the library is used by students. They surveyed 25 students asking them “How many times did you use the library this week?” The results are summarized in the histogram below.

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SAT Math: Statistics and Frequency Tables

Today, we’re going to look at data that is presented in a frequency table. Before we do that,  however, let’s review the different descriptive statistics that you might see on the SAT Math test.

Mean – When people say they are finding the average, what they are usually finding is the mean, which is also called the arithmetic mean. To find the mean, you add up all of the data points and divide by the number of numbers.
Median – The median is the middle number when the numbers are ordered from least to greatest. If you have an even number of numbers, there is no “middle” number. In this case, the median is the mean of the two middle numbers.
Mode – The mode is the number that occurs the most frequently. It is possible to have more than one mode if there is a tie for what occurs most frequently. If each of the numbers appears only once, then we say there is no mode.
Range – The range is the difference between the largest number and the smallest number.

Now, let’s look a few problems that involve data that is presented to you in a frequency table.

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