# III: Expanding and Factoring

Goal: Understand like terms, expanding, and factoring.

This lecture is recorded in two parts!

## Algebraic Expressions

Algebraic expressions are just operations, numbers, and variables mixed together. For example $2x^2 - 3x + 4 \qquad \sqrt{x} + 10 \qquad (x + 1)(x + 2) \qquad (a + b + c)^2(x^2y + 3)$ are all algebraic expressions.

To add and subtract algebraic expressions, look for like terms.

Like terms are terms with the same factors except possibly for the numerical factor (called the coefficient).

Which are like terms?
1. $x^3$ and $3x^3$
2. $-4x^2y^2$ and $3y^2x^2$
3. $-(x+2)(x+1)$ and $3(x+1)(x+2)$

1. Both expressions are $1 \cdot x^3$ and $3\cdot x^3$. $1$ and $3$ are different and both contain the factor $x^3$. They are like terms.
2. Both expressions contain the factors $x^2$ and $y^2$. The coefficient is $-4$ and $3$. They are like terms.
3. These are like terms. Recall that $-a = (-1)\cdot a$. So $\underbrace{(-1)}_{factor}(x+2)(x+1) \qquad$

Remember: identify the factors. This is your guiding light.

To add like terms, we undo the distributive law: $x^3 + 2x^3 = 1\cdot x^3 + 2\cdot x^3 = x^3(1 + 2) = 3x^3$

In essence, add together the coefficients for like terms.

• $x^3 + 2x^2 + 3x^3$
• $(4x^2y + 3xy^2) - (3x^2y^2 - 2xy^2)$
• $2(x+h)^2 - 3 - (2x^2 - 3)$

## Multiplying

Multiplying is also called expanding. Expanding problems are just the distributive law applied.

In expanding problems, you always start with factors and you end with terms (context if not explicitly mentioned is always the entire expression).

Look at the distibutive law: $a\cdot(b + c) = a\cdot b + a \cdot c$

The Law is saying for the factor $a$ outside the parenthesis, distibutive it to each term within the parenthesis.

Expand the following:
• $(2x + 1)(3x - 5)$
• $(x^2 + 2x + 1)(x + 1)$

## Special Product Formulas

Here are three formulas you should be able to recall by heart. Your calculus professor will expect you to immediately be able to see these, especially the first one.

• $(A + B)(A - B) = A^2 - B^2$
• $(A + B)^2 = A^2 + 2AB + B^2$
• $(A - B)^2 = A^2 - 2AB + B^2$

Note that $A$ just needs to be a term. $A$ could be $(x + y)$. It could be $2x$. Hopefully you are starting to see the importance of identifying terms/factors. It is in everything we do!

Expand the following using the formulas:
• $(2x + 1)^2$
• $(2x - 3)(2x + 3)$
• $(x + y - z)(x + y + z)$

## Factoring

Consider this expanding problem: $(x+4)(x+3) = x^2 + 7x + 12$ The expression structure on the left is all factors.

On the right, it is all terms.

If we start from the right and move backwards instead, this process is called factoring.

Factoring converts terms into factors and is the reverse process of expanding.

### Method 1: Greatest Common Factor (among terms)

Identify all terms.

For each term, identify the factors.

From each term, pull out all factors in common among all terms.

Factor each expression:
• $3x^2 - 6x$
• $8x^4y^2 + 6x^3y^3 - 2xy^4$
• $(2x + 4)(x-3) - 5(x-3)$

### Method 2: Factoring the form $ax^2 + bx + c$.

This method works on three terms. Some examples include $x^2 + 4x + 3 \qquad\qquad x^2 - 4 \qquad\qquad 2(x+1)^2 - 7(x+1) + 3$

See this handout for the idea. Let's call this the "new" X method.

Factor:
• $x^2 + 7x + 12$
• $6x^2 + 7x - 5$
• $(5a+1)^2 - 2(5a+1) - 3$

### Method 3: Special Factoring Formulas

The three special product formulas give the special factoring formulas.

• $A^2 - B^2 = (A-B)(A+B)$
• $A^2 + 2AB + B^2 = (A+B)^2$
• $A^2 - 2AB + B^2 = (A-B)^2$
Factor:
• $4x^2 - 25$
• $(x+y)^2 - z^2$

### Method 4: Grouping

This method works on four terms.

We group first two/last two terms together, then use GCF on both groups.

Factor:
• $x^3 +x^2 + 4x + 4$
• $x^3 - 2x^2 - 9x + 18$

The last example shows even though we did have all factors, we sometimes can keep factoring a factor. We may need to use multiple methods.

### Multiple Methods

Remember, before even starting a factoring problem, identify the terms and how many there are.

Doing so helps you to choose a correct strategy.

Factor:
• $2x^4 - 8x^2$
• $3x^2y - 16xy + 5y$