Review of Factoring Techniques (Prerequisites)

TL;DR

Before we dive into solving quadratic equations, it's super helpful to refresh your factoring skills. You'll primarily need to remember how to factor out common terms, factor trinomials, and recognize difference of squares patterns. These techniques let you break down complex expressions into simpler, multiplied parts.

1. The Mental Model

Think of factoring like reverse multiplication. Instead of combining terms (like $2(x+3) = 2x+6$), you're starting with the combined expression ($2x+6$) and trying to find the parts that multiplied together to make it ($2(x+3)$). It's all about finding simpler pieces.

2. The Core Material

Factoring is a key skill for solving quadratics. We'll focus on three main types you'll use constantly.

2.1. Greatest Common Factor (GCF)

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Always look for a GCF first! It simplifies everything. This means finding the largest number and highest power of any variable that divides into every term in the expression.

Example: Factor $6x^2 + 9x$.
Both $6x^2$ and $9x$ share a common factor of $3$ (from the numbers) and $x$ (from the variables). So, the GCF is $3x$.
$6x^2 + 9x = 3x(2x + 3)$

2.2. Factoring Trinomials (like $ax^2 + bx + c$)

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When you have three terms, especially with an $x^2$, $x$, and a constant, you're usually looking to factor it into two binomials, like $(dx+e)(fx+g)$.

2.2.1. Case 1: Leading Coefficient is 1 ($x^2 + bx + c$)

You're looking for two numbers that multiply to give you $c$ and add up to give you $b$.

Example: Factor $x^2 + 7x + 10$.
We need two numbers that multiply to $10$ and add to $7$. These numbers are $2$ and $5$.
So, $x^2 + 7x + 10 = (x+2)(x+5)$

2.2.2. Case 2: Leading Coefficient is NOT 1 ($ax^2 + bx + c$ where $a \neq 1$)

This is a bit trickier, but the 'AC method' (or 'split the middle') is reliable.

1. Multiply $a$ and $c$.
2. Find two numbers that multiply to $ac$ and add to $b$.
3. Rewrite the middle term ($bx$) using these two numbers.
4. Factor by grouping.

Example: Factor $2x^2 + 11x + 12$.
1. $a \times c = 2 \times 12 = 24$.
2. We need two numbers that multiply to $24$ and add to $11$. These are $3$ and $8$.
3. Rewrite the middle term: $2x^2 + 3x + 8x + 12$.
4. Factor by grouping:
$x(2x+3) + 4(2x+3)$
$(x+4)(2x+3)$
So, $2x^2 + 11x + 12 = (x+4)(2x+3)$

2.3. Difference of Squares ($a^2 - b^2$)

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This is a special pattern: if you have one perfect square minus another perfect square, it always factors into a specific form.

$a^2 - b^2 = (a-b)(a+b)$

Example: Factor $x^2 - 49$.
Here, $a=x$ (because $x^2$ is $x$ squared) and $b=7$ (because $49$ is $7$ squared).
So, $x^2 - 49 = (x-7)(x+7)$

Here's a diagram to help visualize the factoring decision process:

graph TD
    Start(("Start Factoring")) --> A{"Is there a GCF?"};
    A -- Yes --> B["Factor out GCF"];
    A -- No --> C{"How many terms are left?"};
    B --> C;
    C -- 2 Terms --> D{"Is it a Difference of Squares? (e.g., $a^2 - b^2$)"};
    D -- Yes --> E["Factor as $(a-b)(a+b)$"];
    D -- No --> F["Can't factor (for now)"];
    C -- 3 Terms --> G{"Is it a trinomial? (e.g., $ax^2 + bx + c$)"};
    G -- Yes --> H{"Is $a=1$?"};
    H -- Yes --> I["Find two numbers that multiply to $c$, add to $b$"];
    H -- No --> J["Use AC Method (Split the middle)"];
    I --> K("Factored Expression");
    J --> K;
    E --> K;
    F --> K;
    C -- 4+ Terms --> L["Try Factoring by Grouping (if not already grouped)"];
    L --> K;

3. Worked Example

Let's factor $3x^2 - 15x - 42$.

1. Look for GCF: All terms are divisible by $3$.
   $3(x^2 - 5x - 14)$
2. Factor the trinomial inside the parentheses: $x^2 - 5x - 14$.
   This is a trinomial where $a=1$. We need two numbers that multiply to $-14$ and add to $-5$.
   The numbers are $-7$ and $2$.
3. Combine the GCF with the factored trinomial:
   $3(x-7)(x+2)$

So, the fully factored expression is $3(x-7)(x+2)$.

4. Key Takeaways

• Always check for a Greatest Common Factor (GCF) first; it simplifies everything.
• For trinomials ($ax^2 + bx + c$), when $a=1$, find two numbers that multiply to $c$ and add to $b$.
• For trinomials where $a \neq 1$, use the AC method (multiply $a \times c$, find factors, split the middle term, then factor by grouping).
• A difference of squares, $a^2 - b^2$, always factors into $(a-b)(a+b)$.
• Factoring is the reverse of distribution; you're finding the factors that multiply to the original expression.
• When factoring, you're looking for common 'pieces' or specific patterns.

Common Mistakes to Avoid:
- Forgetting the GCF; this leads to an incompletely factored expression.
- Mixing up the signs when finding numbers for trinomials (e.g., multiplying to a negative but adding to a positive).
- Trying to factor a sum of squares ($a^2 + b^2$); this doesn't factor over real numbers.
- Not checking your work by multiplying your factored answer back out.

5. Now Try It

Factor the following expressions completely:
1. $4x^3 + 16x^2 - 20x$
2. $x^2 - 11x + 28$
3. $9x^2 - 64$
4. $5x^2 + 13x + 6$

Success looks like you've correctly factored each expression into its simplest multiplicative components. Remember to double-check by multiplying your factors back together to ensure they match the original expression.