Week 5 in Precalc 11

This week in math we learned how to factor polynomial expressions, specifically ‘ugly’ ones using the ‘box method’. A helpful phrase to figure out if a polynomial expression is ugly or not is CDPEU; Can Divers Pee Easily Underwater. What each of these letter actual stand for mathematically are: common difference, difference of squares, pattern, easy, ugly. When you are given a polynomial expression, this expression will help you figure out what type it is.

First, check if the expression has a common difference. For example, 12x+6 has a common difference of 6, so we’d take out the 6 and put it in front, 6(2x+1). Not all expressions will have a common difference, if the expression doesn’t, then move on to the next letter.

Next, is it a difference of squares? A difference of squares will look something like this: $x^2$-25 = (x-5)(x+5). There should be no middle term in the equation if it’s a difference of squares, it should also have a constant and coefficient that are perfect squares. The factored version is conjugates. If it isn’t, move on to the next letter once again.

Does it have the pattern? If it has ‘the’ pattern it will be in this form: $x^2$+#x+#. As long as there is no number in front of the x, this makes it an easy expression. Which, is our next letter.

If it has this pattern and doesn’t have a number in front of the first term, then it as an easy polynomial and we can solve it fairly quickly. First, you have to find all the number that multiply to the constant. Whichever pair adds to the middle term will be the second terms after x in the factored version. For example,

$x^2$+8x+16 (numbers that multiply to 16: 1·16, 2·8, 4·4. Which two add to 8? 4 and 4)

So, the factored version is (x+4)(x+4).

If none of these so far apply to the expression, then it may be an ugly polynomial. There are many different ways to solve one of these expressions, but I’ll explain the ‘box method’. Refer to the example below as I explain the steps. First, draw out a four square box. In the first box put the first term and in the last box put the last term. Multiply these terms by each other and find all the numbers that multiply into that number. Whichever numbers that also add up to the middle term are the two numbers that you’ll attach to x and put into the second and third squares of the box. Go from the top left box and look across to find the number that both of those squares have in common and write it beside the square. Do this for all other 3 rows. The numbers on the left side will be inside one bracket and the ones on the other side will be inside the next brackets. There is your answer. You can check your answer by multiplying it out back into its original form to see if it’s correct. Follow these steps as you look at the example below.