Category Archives: MATH 86/163

Rational Expressions and Equations

What is a Rational Expression?

A rational number is a number that can be written as a quotient of integers.  In other words, it’s any number that can create a nice and neat fraction. A rational expression is also a quotient, it’s just made up of polynomials. A rational expression can be written in the form P/Q. For example:

Evaluating Rational Expressions

Rational expressions have different numerical values depending on what values replace the variables. For example:

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Factoring Polynomials

Finding the Greatest Common Factor (GCF)

Finding the greatest common factor simply involves finding the largest number or term which will fit evenly into each number or term in a list. The way I like to go about this is by breaking each number or term into its smallest parts. Break each number down until you are multiplying together only prime numbers. All the numbers that they have in common should then be multiplied back up to create the GCF.

This is what it would look like in a list of numbers:

And this is what it would look like in a list of terms:

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Introduction to Exponents and Polynomials

Evaluating Exponential Expressions

When working with exponents, it might be more helpful to think of them as multiple instances of multiplication. Some exponents are going to be more straight-forward, but be careful of the writing of some exponents.

Let’s take a look at some examples of evaluating exponential expressions:

In the expression below, this is an illustration of what we mean when we say that an exponent is like multiple multiplications. The exponents signify the number of times that the number 2 should be multiplied by itself.

In the next expression, the -3 is in parentheses. This means that the exponent outside of the parentheses needs to be applied to the number as a whole, including its being negative.

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Statistics: Discrete Probability Distributions

Introduction to Discrete Probability Distributions

Discrete versus Continuous Variables

A discrete variable typically originates from a counting process while a continuous variable usually comes from a measuring process. An easy way to make the distinction between a discrete and a continuous variable is that discrete variables are usually whole numbers with no decimals. Continuous variables on the other hand frequently take the form of decimals. For instance, the number of people which exist within a group is a discrete variable because it’s always a whole number, while a person’s weight would be continuous since it can typically be measured to multiple decimal places.

The Probability Distribution for a Discrete Variable

A probability distribution for a discrete variable is simply a compilation of all the range of possible outcomes and the probability associated with each possible outcome. Since, probability in general, by definition, must sum to 1, the summation of all the possible outcomes must sum to 1. For example, if you’re flipping a coin once, there’s a 1 in 2 chance it will land on heads, and a 1 in 2 chance it will land on tails; 1/2 + 1/2 = 1. In this way, measuring probability is similar to the use of percentages. Percentages are always measured out of 100 while probability is always measured out of 1. This is true for all probability measurements Continue reading