Congratulations, you've just finished the first piece of this Udacity course. Now that we've talked about a number of new ideas, let's take a few minutes to summerize these topics. We've talked about classification of numbers, algebraic terminoligy, simplification of expressions, and subsitution. Let's try a few problems to review. Let's think back to our world of numbers. This picture might look familiar to you, but notice that it doesn't have the names of the sets of numbers we've talked about on it. Please take a minute to match the letters with the appropriate sets.
So here is your solution. Take a second to look at the placement of each name, and recall how this diagram actually works. Any circle that's contained within another circle is a subset of that larger set. For example, natural numbers are a subset of the whole numbers. This means that if a number belongs in a certain category it also belongs in every category that surrounds that circle. So all natural numbers are whole numbers, and integers, and rational numbers, and real numbers.
Now that our world of numbers is labeled with categories, let's put some numbers in it. Here are a bunch of numbers. Type the letter that corresponds to each number in the proper box in the diagram. Remember, that on the diagram, numbers are positioned in the smallest set they belong in or is far into the graph as they can be placed.
Please take a minute to look at the placement of each letter. There are a few numbers you may have wanted to place in several positions. For example, you may have been tempted to write 3 as an integer, as a whole number, and of course a natural number. However, the correct placement on the graph is for the number to be placed in the smallest circle it can be placed in. Remember, by placing a number in the smallest circle, you are placing it in all of the larger circles that surround the smallest circle.
Please look at the problem below and put it in its most simplified form.
So, you might have noticed that x^2 and And for now, we're going to leave the rest of the expression as it is. So now, the next thing I noticed with this expression is that we need to multiply the quantity x+2 times 7x^2-1. This gives us 7x^3-x+14x^2-2, and we'll leave the rest of the expression the same. So, our next step is to distribute the 4 with everything inside the parenthesis. When we do this, inside the parenthesis, we get 28x^3-4x+56x^2-8 and we'll leave the rest of the expression the same for now. So, our next step is to distribute the -1 in front of the parenthesis with everything inside and combine like terms. This gives us our solution of -28x^3-56x^2+7x+1.
Now, let's review substitution. Recall the last problem we did, -28x^3-56x^2+7x+1. Please substitute in -2 for x and evaluate the expression for -2.
So the first step is to substitute a -2 for x throughout the expression. The next step is to just simplify 224 - 224 - 13, leaving us with the solution of -13.