## Day 1

- Topic of the week: What is factoring? Why do we care? How do we do it?
- Distributing binomials. Write each of these in an equivalent form:

\begin{align*}
(x+2)(x+3) & \\
(x+4)(x-1) & \\
(x-5)(x-3) \\
\end{align*}

- What patterns do you notice in the work from 1?

\begin{align*}
(x+2)(x+3) & \\
x^2 + 2x + 3x + 6 & \\
x^2 + 5x + 6 &
\end{align*}

If you write each of the distributing steps, every single time there is a part in the middle that is the sum of the two numbers in the two starting binomials, and the last number is the product of the two numbers in the binomial. \(2+3 = 5\) and \(2\cdot 3 = 6\). Does that pattern always hold?

\begin{align*}
(2x+5)(x+3) & \\
2x^2 + 6x + 5x + 15
\end{align*}

These have a more complicated form, but the last term is still the product of the last two numbers \(5\cdot 3 = 15\), the first term is still the product of the first terms in the binomials \(2x \ cdot x = 2x^2\), and the middle terms are from adding together something.

## Day 2

- Write each of these expressions in a a form that is equivalent to the original:

\begin{align*}
6 \cdot 84 & \\
3(a^2 + 9) & \\
4x(2x - 5) & \\
-3x(5x - 3x) & \\
\end{align*}

- Each of these is distributing. How is the first one distribution?

\begin{align*}
6 \cdot 84 & \\
6 (80 + 4) & \\
480 + 24 & \\
504 &
\end{align*}

- They are all making use of the distributive property. Mental math in general makes frequent use of the distributive property to make the substeps of a calculation simpler.
- Now how about writing the number 492 as a multiplication problem:
There are a lot of ways to do it, but they all involve indentifying numbers that divide evenly into 492. Like \(2 \cdot 246 = 496\), a skill we will be practicing a lot in the next few days.

- Distribute the following, writing out all your steps, then think about what steps you would follow to
reversethe process:

\begin{align*}
5x(2x - 7) & \\
10x^2 - 35x & \\
5(x^2 - 7x) & \\
5x(x - 7) &
\end{align*}

## Day 3

- Write the following as multiplication problems (ie,
reverse distribute):

\begin{align*}
1.& 25b^2 - 35 \\
2.& x^2 - 2x \\
3.& -27x^2 - 9x
\end{align*}

- You have to identify numbers that divide evenly into each term in the original expression, and pay attention to the sign of each term.
- Now distribute these, and think about the process to reverse distribute:

\begin{align*}
(x+1)(x+2) & \\
x^2 + 2x + 1x + 2 & \\
x^2 + 3x + 2 &
\end{align*}

What do you need to do/know to be able to reverse this? The un-distributed form has the numbers 1 and 2 in the binomial expressions. Those numbersaddto get 3 andmultiplyto get 2. So it seems we are using the factors of 2, which also add up to 3.

\begin{align*}
x^2 + 3x + 2 & \\
x^2 + 2x + 1x + 2 & \\
(x^2 + 2x) + (1x + 2) &\\
x(x + 2) + 1(x+2) &
\end{align*}

Notice that this last form has the same factor in each term (\((x+2)\)). Since this number, whatever it is, is in both terms, we can factoritout of the expression:

\begin{align*}
x(x+2) + 1(x+2) &\\
(x+2)(x + 1) &
\end{align*}

That last step is probably the one that will make the least sense, so try to make sure you understand where it came from before you continue.

- Now try these, trying to un-distribute or factor them:

\begin{align*}
1. & x^2 + 5x + 6 \\
2. & x^2 + 7x + 10 \\
3. & x^2 + 10x + 16
\end{align*}

- Why do we care, again? Graph each of the expressions in the preceding question as equations (\(y= x^2 + 5x +6\), \(y = x^2 + 7x + 10\), and \(y=x^2 + 10x+16\), and compare the graph to the factored form of the expressions. Why do we care about factoring? What does the factored form reveal, that is not apparent by looking at the general form of the equation (\(y = ax^2 + bx + c\) or the vertex form (\(y - h = a(x-k)^2\)?
- Different ways of expressing the same information make different features more or less apparent.

## Day 4

- Practice, practice, with different types of quadratic equations involving different complexities in the factoring process.
- Ask Mark for practice problems if you want to do more of these -- this is a key skill that is involved in a lot of other things we will do this year, and in many other math classes. It's a good one to practice a lot, if you think you need it.

Is something unclear? Leave a comment below:

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