Study Notes: Factorization

Welcome to the World of Factorization!

Hello! Get ready to learn one of the most useful skills in algebra: factorization.

"What is factorization?" you might ask. Think of it like being a detective. When you expand an expression like $$ 2(x + 5) $$ to get $$ 2x + 10 $$, you're mixing ingredients to bake a cake. Factorization is the reverse! You start with the finished cake ($$ 2x + 10 $$) and figure out the original ingredients ($$ 2 $$ and $$ (x + 5) $$).

It's basically the process of "un-multiplying" an expression. This skill is super important because it helps us simplify complicated expressions and solve equations more easily. Don't worry if it sounds tricky at first – we'll break it down step-by-step!

What is Factorization? The Big Idea

The Reverse of Expansion

You already know how to expand expressions by multiplying out brackets. For example:
Expansion: $$ 3x(x - 4) \rightarrow 3x^2 - 12x $$

Factorization does the exact opposite. It takes the result and puts it back into its "factors" (the things that were multiplied together).
Factorization: $$ 3x^2 - 12x \rightarrow 3x(x - 4) $$

The expression $$ 3x^2 - 12x $$ is written as a product of its factors, which are $$ 3x $$ and $$ (x - 4) $$.

Key Takeaway

Factorization is the process of writing an algebraic expression as a product of two or more simpler expressions (its factors).

Method 1: Taking Out the Common Factor

Finding the Greatest Common Factor (GCF)

This is the first thing you should always look for in any factorization problem. It's often the easiest first step! A common factor is a number or variable that can be divided into every single term in the expression. We want to find the Greatest Common Factor (GCF).

Here's the step-by-step process:

1. Look at all the terms in the expression.
2. Find the largest number that divides into all the coefficients (the number parts).
3. Look for any variables that appear in all terms. Take the one with the lowest power.
4. Write this GCF outside a new set of brackets.
5. To find what goes inside the brackets, divide each original term by the GCF.

Example 1: Simple Numbers

Factorise $$ 5x + 15 $$
1. The terms are $$ 5x $$ and $$ 15 $$.
2. The largest number that divides into both 5 and 15 is 5.
3. The variable $$x$$ is only in the first term, not the second, so it's not a common factor.
4. The GCF is 5. We write it outside a bracket: $$ 5( \quad ) $$
5. Divide the original terms by 5:
     $$ 5x \div 5 = x $$
     $$ 15 \div 5 = 3 $$
So, what goes inside the bracket is $$ x + 3 $$.
Answer: $$ 5(x+3) $$

Example 2: With Variables

Factorise $$ 6a^2 + 9ab $$
1. The terms are $$ 6a^2 $$ and $$ 9ab $$.
2. The GCF of the numbers 6 and 9 is 3.
3. The variable 'a' is in both terms. The lowest power is $$ a^1 $$ (or just $$ a $$). So, 'a' is part of the GCF.
4. The GCF is 3a. We write: $$ 3a( \quad ) $$
5. Divide:
     $$ 6a^2 \div 3a = 2a $$
     $$ 9ab \div 3a = 3b $$
Answer: $$ 3a(2a + 3b) $$

Common Mistake Alert!

When a term is identical to the GCF, don't forget the '1'!
Factorise $$ 7y + 7 $$. The GCF is 7.
Correct: $$ 7(y+1) $$. (Because $$ 7 \div 7 = 1 $$)
Incorrect: $$ 7(y) $$

Key Takeaway

Always, always, ALWAYS look for a common factor first! It makes every other step easier.

Method 2: Factorization by Grouping

When there's no single common factor

What if you have an expression with four terms, and there's no factor common to all of them? Don't panic! It might be time for grouping.

Here's how to do it (Team Up!):

1. Arrange the expression into two pairs (groups).
2. Factor out the common factor from the first pair.
3. Factor out the common factor from the second pair.
4. You should now have an identical bracket in both parts. This bracket is your new common factor!
5. Factor out the common bracket.

Step-by-Step Example

Factorise $$ xy + 2x + 3y + 6 $$
1. There is no factor common to all four terms. So, let's group them:
$$ (xy + 2x) + (3y + 6) $$

2. Factor the first pair. The common factor in $$ (xy + 2x) $$ is $$ x $$.
$$ x(y + 2) $$

3. Factor the second pair. The common factor in $$ (3y + 6) $$ is $$ 3 $$.
$$ +3(y + 2) $$

4. Now put it together: $$ x(y + 2) + 3(y + 2) $$. Look! The bracket $$ (y + 2) $$ is a common factor.

5. Factor out the common bracket $$ (y+2) $$. What's left over? The '$$x$$' and the '$$+3$$'. These form the second bracket.
Answer: $$ (y+2)(x+3) $$

Key Takeaway

If you see four terms, think about using the grouping method. The goal is to create a common bracket.

Method 3: Using Identities (The Super Shortcuts!)

Some expressions follow special patterns called identities. If you can spot these patterns, you can factorize them in seconds!

Identity 1: Difference of Two Squares (DOTS)

Look for this pattern: (Something squared) - (Another thing squared). It must have two terms, a minus sign between them, and both terms must be perfect squares.

The formula is: $$ \mathbf{a^2 - b^2 = (a - b)(a + b)} $$

Example

Factorise $$ x^2 - 16 $$
1. Is the first term a square? Yes, $$ x^2 $$ is $$ (x)^2 $$. So, $$ a = x $$.
2. Is the second term a square? Yes, $$ 16 $$ is $$ (4)^2 $$. So, $$ b = 4 $$.
3. Is there a minus sign? Yes.
4. It's a perfect match for DOTS! Just plug $$ a $$ and $$ b $$ into the formula $$ (a - b)(a + b) $$.
Answer: $$ (x - 4)(x + 4) $$

Identity 2: Perfect Square Trinomials

These have three terms (which is why they're called trinomials). Look for these two patterns:

Pattern 1: $$ \mathbf{a^2 + 2ab + b^2 = (a + b)^2} $$
Pattern 2: $$ \mathbf{a^2 - 2ab + b^2 = (a - b)^2} $$

How to check if it's a Perfect Square Trinomial:

1. Is the first term a perfect square? If yes, find '$$a$$'.
2. Is the last term a perfect square? If yes, find '$$b$$'.
3. Is the middle term equal to $$ 2 \times a \times b $$ (ignoring the sign for a moment)?
4. If you answered YES to all three, it's a perfect square! The sign in your bracket will be the same as the sign of the middle term.

Example

Factorise $$ x^2 + 10x + 25 $$
1. First term is $$ x^2 $$, which is $$ (x)^2 $$. So, $$ a = x $$.
2. Last term is $$ 25 $$, which is $$ (5)^2 $$. So, $$ b = 5 $$.
3. Let's check the middle term. Is it $$ 2ab $$?
$$ 2 \times x \times 5 = 10x $$. Yes, it matches!
4. The middle term has a '+' sign, so we use the $$ (a+b)^2 $$ formula.
Answer: $$ (x + 5)^2 $$

Key Takeaway

Learning to spot these identities is like having a superpower. Always look for perfect squares at the beginning and end of a three-term expression, or for a difference of two squares in a two-term expression.

Method 4: The Cross-Method

For Trinomials like $$ ax^2 + bx + c $$

When a trinomial (3 terms) isn't a perfect square, the cross-method is your best friend. It's a way to find the two brackets by solving a small puzzle.

Step-by-Step with an Example

Factorise $$ x^2 + 7x + 12 $$
The Puzzle: We need two numbers that multiply to give the last term (+12) and add to give the middle term's coefficient (+7).

1. Draw a cross. On the left, write the factors of the first term ($$x^2$$). This is usually just $$ x $$ and $$ x $$.
$$ \begin{matrix} x \\ & \Large{\times} \\ x \end{matrix} $$
2. List factors of the last term. On the right, we need pairs of numbers that multiply to 12.
Pairs for 12: (1, 12), (2, 6), (3, 4)

3. Test the pairs. We place a pair on the right side of the cross, then cross-multiply and add the results. Our goal is to get the middle term, $$ 7x $$. Let's try (3, 4).
$$ \begin{array}{ccc} x & & +3 \\ & \Large{\times} & \\ x & & +4 \end{array} $$
Cross-multiply: $$ (x \times 4) = 4x $$ and $$ (x \times 3) = 3x $$.
Add them: $$ 4x + 3x = 7x $$. It's a match!

4. Write the answer. Read across horizontally to get your brackets.
The top row gives the first bracket: $$ (x + 3) $$
The bottom row gives the second bracket: $$ (x + 4) $$
Answer: $$ (x+3)(x+4) $$

Did you know?

The cross-method is a visual way of reversing the FOIL method (First, Outer, Inner, Last) that you use for expanding brackets. The two cross-products you calculate are the 'Outer' and 'Inner' parts of FOIL!

Key Takeaway

The cross-method is a reliable tool for factoring trinomials. The key is to test pairs of factors for the last term until the cross-products add up to the middle term. Practice is the best way to get fast at this!

Chapter Summary & Your Factorization Strategy

A Checklist for Success

Feeling overwhelmed by all the methods? Don't be! Just follow this simple checklist every time you need to factorise an expression.

Step 1: GCF First!
Does the expression have a Greatest Common Factor? If yes, factor it out immediately.

Step 2: Count the Terms
After taking out the GCF, how many terms are left inside the bracket?

    - If there are 2 terms, check if it's a Difference of Two Squares ($$a^2 - b^2$$).
    - If there are 3 terms, check if it's a Perfect Square Trinomial ($$a^2 \pm 2ab + b^2$$). If not, use the Cross-Method.
    - If there are 4 terms, try Factorization by Grouping.

Step 3: Check Your Answer
You can always check your final answer by expanding it. If you get the original expression, you know you're correct!

Final Encouragement

Well done! You've just learned the fundamentals of factorization. This is a huge step in your algebra journey. Like any skill, the more you practice these methods, the easier and faster you will become. Keep up the great work!