Factoring Polynomials

Part 1: Greatest Common Factor (GCF) 🔍

Welcome to factoring! We'll start with the most important rule:

Always Look for the GCF First!

The Greatest Common Factor (GCF) is the largest expression that divides evenly into all terms.

Why start here? Because factoring out the GCF first makes everything else easier! It's like simplifying before you solve - and you should always check for a GCF before trying any other factoring technique.

What You'll Learn in Part 1:

• 🔍 How to identify the GCF
• 📝 How to factor out the GCF
• ✅ Practice problems to master the technique

How to Find and Factor Out the GCF 🎯

Step 1: Identify the GCF

• Look at the coefficients - What's the largest number that divides all of them?
• Look at the variables - Take the lowest power of each variable that appears in all terms

Step 2: Factor Out the GCF This means you're doing two things simultaneously:

1. Divide each term by the GCF
2. Write the GCF outside parentheses with the divided terms inside

The Process: $\text{Original} \rightarrow \text{GCF}(\text{each term} \div \text{GCF})$

Let's see this in action with examples...

Example 1: Factoring Out the GCF 📝

Problem: $6x^3 + 9x^2$

Step 1: Find the GCF

• Coefficients: GCF of 6 and 9 is 3
• Variables: Both terms have $x$, lowest power is $x^2$
• GCF = $3x^2$

Step 2: Factor Out the GCF

Divide each term by $3x^2$ and write $3x^2$ outside:

$6x^3 + 9x^2 = 3x^2(\text{?} + \text{?})$

Divide each term:

• $6x^3 \div 3x^2 = 2x$ ← First term inside parentheses
• $9x^2 \div 3x^2 = 3$ ← Second term inside parentheses

Final Answer: $6x^3 + 9x^2 = 3x^2(2x + 3)$ ✅

Check your work: Multiply back out: $3x^2 \cdot 2x + 3x^2 \cdot 3 = 6x^3 + 9x^2$ ✓

Example 2: More Complex GCF 📝

Problem: $10x^4y^2 - 15x^2y^3 + 5xy$

Step 1: Find the GCF

• Coefficients: GCF of 10, 15, and 5 is 5
• Variables: All have $x$ (lowest: $x$) and $y$ (lowest: $y$)
• GCF = $5xy$

Step 2: Factor Out the GCF

Divide each term by $5xy$ and write $5xy$ outside:

$10x^4y^2 - 15x^2y^3 + 5xy = 5xy(\text{?} - \text{?} + \text{?})$

Divide each term:

• $10x^4y^2 \div 5xy = 2x^3y$ ← First term
• $15x^2y^3 \div 5xy = 3xy^2$ ← Second term
• $5xy \div 5xy = 1$ ← Third term

Final Answer: $10x^4y^2 - 15x^2y^3 + 5xy = 5xy(2x^3y - 3xy^2 + 1)$ ✅

Important: Don't forget the "+ 1" at the end! When a term equals the GCF exactly, dividing gives you 1.

Practice: Identify the GCF 🎯

First, let's practice just identifying the GCF (you don't need to factor it out yet).

For each polynomial, determine what the Greatest Common Factor is.

You need to answer 3 questions correctly in a row to continue.

Practice: Factor Out the GCF 🎯

Now it's your turn to factor completely!

For each polynomial:

1. Find the GCF
2. Divide each term by the GCF
3. Write the GCF outside with the divided terms inside

You need to answer 3 questions correctly in a row to complete Part 1.

Part 1 Complete! Great Job! 🎉

You've mastered finding and factoring out the GCF!

Remember:

• ✅ Always check for a GCF first - it's the foundation of all factoring
• ✅ Look at both coefficients and variables
• ✅ Take the lowest power of each variable that appears in all terms

Next Up: Part 2 - Difference of Squares

In the next part, you'll learn one of the most powerful factoring patterns: $a^2 - b^2 = (a + b)(a - b)$

Ready to continue? Click "Next" or use the navigation menu above!

Part 2: Difference of Squares 📊

You've mastered the GCF! Now let's learn one of the most important patterns in algebra.

The Difference of Squares Pattern

This is a special pattern you'll use throughout algebra and calculus!

The Pattern: $a^2 - b^2 = (a + b)(a - b)$

What makes it special:

• ✅ It's fast and easy to recognize
• ✅ It always works the same way
• ✅ You'll see it everywhere in math!

How to Recognize the Difference of Squares 🎯

Three Requirements:

1. ✅ Two terms only (nothing else!)
2. ✅ Both terms are perfect squares
3. ✅ Subtraction sign between them (difference, not sum!)

The Pattern: $a^2 - b^2 = (a + b)(a - b)$

The Method:

1. Identify what's being squared in the first term → this is "$a$"
2. Identify what's being squared in the second term → this is "$b$"
3. Write two factors: $(a + b)(a - b)$

Key Insight: One factor has addition, one has subtraction!

Let's see this in action with some examples...

Example 1: Basic Difference of Squares 📝

Problem: Factor $x^2 - 25$

Step 1: Check the three requirements

• ✅ Two terms only: $x^2$ and $25$
• ✅ Both are perfect squares: $x^2 = (x)^2$ and $25 = (5)^2$
• ✅ Subtraction sign between them

Step 2: Identify $a$ and $b$

• $a = x$ (because $x^2 = (x)^2$)
• $b = 5$ (because $25 = (5)^2$)

Step 3: Apply the pattern $a^2 - b^2 = (a + b)(a - b)$ $x^2 - 25 = (x + 5)(x - 5)$

Verification: Let's check by expanding $(x + 5)(x - 5) = x^2 - 5x + 5x - 25 = x^2 - 25$ ✅

Example 2: Coefficients on Variables 📝

Problem: Factor $9x^2 - 16$

Step 1: Check the three requirements

• ✅ Two terms only: $9x^2$ and $16$
• ✅ Both are perfect squares: $9x^2 = (3x)^2$ and $16 = (4)^2$
• ✅ Subtraction sign between them

Step 2: Identify $a$ and $b$

• $a = 3x$ (because $9x^2 = (3x)^2$)
• $b = 4$ (because $16 = (4)^2$)

Step 3: Apply the pattern $a^2 - b^2 = (a + b)(a - b)$ $9x^2 - 16 = (3x + 4)(3x - 4)$

Verification: Let's check by expanding $(3x + 4)(3x - 4) = 9x^2 - 12x + 12x - 16 = 9x^2 - 16$ ✅

Example 3: Higher Powers and Multiple Variables 📝

Problem: Factor $49x^4 - 64y^2$

Step 1: Check the three requirements

• ✅ Two terms only: $49x^4$ and $64y^2$
• ✅ Both are perfect squares: $49x^4 = (7x^2)^2$ and $64y^2 = (8y)^2$
• ✅ Subtraction sign between them

Step 2: Identify $a$ and $b$

• $a = 7x^2$ (because $49x^4 = (7x^2)^2$)
• $b = 8y$ (because $64y^2 = (8y)^2$)

Step 3: Apply the pattern $a^2 - b^2 = (a + b)(a - b)$ $49x^4 - 64y^2 = (7x^2 + 8y)(7x^2 - 8y)$

Verification: Let's check by expanding $(7x^2 + 8y)(7x^2 - 8y) = 49x^4 - 56x^2y + 56x^2y - 64y^2 = 49x^4 - 64y^2$ ✅

Common Mistake to Avoid! ⚠️

IMPORTANT: The difference of squares pattern ONLY works with subtraction!

Does NOT Factor (with real numbers): $x^2 + 25$

This is a sum of squares, not a difference. It cannot be factored using real numbers.

Why? If we try to use a similar pattern:

• $(x + 5)(x + 5) = x^2 + 10x + 25$ ❌ (has a middle term)
• $(x - 5)(x - 5) = x^2 - 10x + 25$ ❌ (has a middle term)
• $(x + 5)(x - 5) = x^2 - 25$ ❌ (this is difference, not sum!)

Rule: $a^2 + b^2$ is prime (cannot be factored with real numbers)

Pro Tip: Sometimes you need to factor out a GCF first to reveal a difference of squares!

Example: Factor $2x^2 - 50$

1. Factor out the GCF first: $2(x^2 - 25)$
2. Now apply difference of squares: $2(x + 5)(x - 5)$ ✅

Always check for a GCF before applying any factoring pattern!

Practice: Difference of Squares 🎯

Factor each expression using the difference of squares pattern.

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

You need to answer 3 questions correctly in a row to proceed to the Mini-Boss Challenge!

Mini-Boss Challenge ⚔️

Test your skills against the Factoring Guardian!

Part 2 Complete! Excellent Work! 🎉

You've mastered the difference of squares pattern AND defeated the Factoring Guardian!

Remember:

• ✅ $a^2 - b^2 = (a + b)(a - b)$
• ✅ Look for two perfect squares with subtraction
• ✅ Always check for a GCF first!

Next Up: Part 3 - Simple Trinomials

In the next part, you'll learn how to factor trinomials like $x^2 + 7x + 12$ where the leading coefficient is 1.

Ready to continue? Click "Next" or use the navigation menu above!

Part 3: Simple Trinomials 🎯

Great progress! Now let's tackle trinomials (three-term polynomials).

Factoring Simple Trinomials

Simple trinomials have the form: $x^2 + bx + c$ where the leading coefficient is 1.

The Goal: Find two numbers that multiply and add to specific values!

What You'll Learn:

• 🎯 The "multiply and add" strategy
• ➕➖ How to handle positive and negative signs
• ✅ Practice to build speed and confidence

The Multiply and Add Strategy 🎯

For trinomials in the form: $x^2 + bx + c$

The Pattern: $x^2 + bx + c = (x + m)(x + n)$

The Two Rules:

1. Find two numbers where: $m + n = b$ (they add to the middle coefficient)
2. And: $m \times n = c$ (they multiply to the constant)

Strategy:

1. List all factor pairs of $c$
2. Find which pair adds to $b$
3. Write as $(x + m)(x + n)$

Sign Rules to Remember:

• If $c$ is positive → both numbers have the same sign
• If $c$ is negative → numbers have opposite signs
• The sign of $b$ tells you which number is larger

Let's see this in action...

Example 1: Both Numbers Positive 📝

Problem: Factor $x^2 + 7x + 12$

Step 1: Identify what we need

• Need $m + n = 7$ (add to the middle coefficient)
• Need $m \times n = 12$ (multiply to the constant)
• Since $c = 12$ is positive, both numbers have the same sign
• Since $b = 7$ is positive, both numbers are positive

Step 2: List factor pairs of 12

• $1 \times 12 = 12$ → $1 + 12 = 13$ ❌
• $2 \times 6 = 12$ → $2 + 6 = 8$ ❌
• $3 \times 4 = 12$ → $3 + 4 = 7$ ✅

Step 3: Write the factored form $x^2 + 7x + 12 = (x + 3)(x + 4)$

Verification: $(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$ ✅

Example 2: Both Numbers Negative 📝

Problem: Factor $x^2 - 5x + 6$

Step 1: Identify what we need

• Need $m + n = -5$ (add to the middle coefficient)
• Need $m \times n = 6$ (multiply to the constant)
• Since $c = 6$ is positive, both numbers have the same sign
• Since $b = -5$ is negative, both numbers are negative

Step 2: List negative factor pairs of 6

• $(-1) \times (-6) = 6$ → $-1 + (-6) = -7$ ❌
• $(-2) \times (-3) = 6$ → $-2 + (-3) = -5$ ✅

Step 3: Write the factored form $x^2 - 5x + 6 = (x - 2)(x - 3)$

Verification: $(x - 2)(x - 3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6$ ✅

Example 3: Opposite Signs 📝

Problem: Factor $x^2 + 2x - 15$

Step 1: Identify what we need

• Need $m + n = 2$ (add to the middle coefficient)
• Need $m \times n = -15$ (multiply to the constant)
• Since $c = -15$ is negative, numbers have opposite signs
• Since $b = 2$ is positive, the positive number is larger

Step 2: List factor pairs of 15 with opposite signs

• $1 \times (-15) = -15$ → $1 + (-15) = -14$ ❌
• $(-1) \times 15 = -15$ → $-1 + 15 = 14$ ❌
• $3 \times (-5) = -15$ → $3 + (-5) = -2$ ❌
• $(-3) \times 5 = -15$ → $-3 + 5 = 2$ ✅

Step 3: Write the factored form $x^2 + 2x - 15 = (x - 3)(x + 5)$

Verification: $(x - 3)(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15$ ✅

Common Mistakes to Avoid! ⚠️

Mistake 1: Forgetting to Check Both Add AND Multiply

• ❌ For $x^2 + 8x + 12$, seeing $2 + 6 = 8$ and writing $(x + 2)(x + 6)$
• Problem: $2 \times 6 = 12$ ✅ but this gives $x^2 + 8x + 12$ which is correct!
• Actually, for $x^2 + 7x + 12$: need $3 \times 4 = 12$ AND $3 + 4 = 7$

Mistake 2: Wrong Signs

• ❌ For $x^2 - 7x + 12$, writing $(x + 3)(x + 4)$
• Problem: This gives $x^2 + 7x + 12$ (positive middle term!)
• ✅ Correct: $(x - 3)(x - 4) = x^2 - 7x + 12$

Mistake 3: Not Checking for GCF First

• ❌ Trying to factor $2x^2 + 14x + 24$ directly
• ✅ Factor out GCF first: $2(x^2 + 7x + 12) = 2(x + 3)(x + 4)$
• Always check for a GCF before using other methods!

Mistake 4: Mixing Up Which Number Goes Where

• For $x^2 + 2x - 15$: found $5$ and $-3$
• ❌ Writing $(x + 3)(x - 5)$ (backwards!)
• ✅ Correct: $(x + 5)(x - 3)$
• Tip: FOIL your answer to check!

Pro Tip: Always verify by expanding your answer! If it doesn't match the original, try again.

Practice: Simple Trinomials 🎯

Factor each trinomial where the leading coefficient is 1.

Find two numbers that multiply to $c$ and add to $b$.

You need to answer 4 questions correctly in a row to proceed to the Mini-Boss!

Mini-Boss Challenge ⚔️

Face The Polynomial Warrior!

Part 3 Complete! You're on Fire! 🔥

You've mastered factoring simple trinomials AND defeated The Polynomial Warrior!

Remember:

• ✅ For $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$
• ✅ If $c$ is positive, both numbers have the same sign
• ✅ If $c$ is negative, the numbers have opposite signs
• ✅ Always check for a GCF first!

Next Up: Part 4 - Complex Trinomials

In the next part, you'll learn the AC Method for factoring trinomials when the leading coefficient is NOT 1 (like $2x^2 + 7x + 3$).

Ready for the challenge? Click "Next" or use the navigation menu!

Part 4: Complex Trinomials (AC Method) 🚀

You're doing great! Now for the trickier trinomials.

When the Leading Coefficient is NOT 1

Complex trinomials have the form: $ax^2 + bx + c$ where $a \neq 1$.

Examples: $2x^2 + 7x + 3$ or $3x^2 - 10x + 8$

The Solution: We'll use the AC Method (also called grouping) - a systematic approach that always works!

What You'll Learn:

• 📐 The AC Method step-by-step
• 🔄 How to factor by grouping
• ✅ Practice with challenging trinomials

The AC Method (Factoring by Grouping) 🚀

For trinomials: $ax^2 + bx + c$ where $a \neq 1$

The 4-Step Process:

Step 1: Multiply $a \times c$ (the first and last coefficients)

Step 2: Find two numbers that multiply to $ac$ and add to $b$

Step 3: Rewrite the middle term using these two numbers

Step 4: Factor by grouping

• Group the first two terms and last two terms
• Factor out the GCF from each pair
• Factor out the common binomial

Pro Tip: In Step 4, both parentheses should be identical. If they're not, you made an error!

Sign Rules:

• If $c$ is positive: both numbers have the same sign as $b$
• If $c$ is negative: numbers have opposite signs, larger one matches sign of $b$

Example 1: Both Positive ✨

Factor: $2x^2 + 7x + 3$

Step 1: Multiply $a \times c$ $ac = 2 \times 3 = 6$

Step 2: Find two numbers that multiply to 6 and add to 7

• Try: 1 and 6 → 1 + 6 = 7 ✅ and 1 × 6 = 6 ✅
• Perfect!

Step 3: Rewrite the middle term $2x^2 + 7x + 3 = 2x^2 + 1x + 6x + 3$

Step 4: Factor by grouping $= (2x^2 + 1x) + (6x + 3)$ $= x(2x + 1) + 3(2x + 1)$ $= (x + 3)(2x + 1)$ ✅

Check: $(x + 3)(2x + 1) = 2x^2 + x + 6x + 3 = 2x^2 + 7x + 3$ ✅

Example 2: Negative Middle Term 🔄

Factor: $3x^2 - 10x + 8$

Step 1: Multiply $a \times c$ $ac = 3 \times 8 = 24$

Step 2: Find two numbers that multiply to 24 and add to -10

• Need both negative (since $c$ is positive but $b$ is negative)
• Try: -4 and -6 → -4 + (-6) = -10 ✅ and (-4) × (-6) = 24 ✅

Step 3: Rewrite the middle term $3x^2 - 10x + 8 = 3x^2 - 4x - 6x + 8$

Step 4: Factor by grouping $= (3x^2 - 4x) + (-6x + 8)$ $= x(3x - 4) - 2(3x - 4)$ $= (x - 2)(3x - 4)$ ✅

Check: $(x - 2)(3x - 4) = 3x^2 - 4x - 6x + 8 = 3x^2 - 10x + 8$ ✅

Example 3: Negative Last Term ⚡

Factor: $2x^2 + 5x - 12$

Step 1: Multiply $a \times c$ $ac = 2 \times (-12) = -24$

Step 2: Find two numbers that multiply to -24 and add to 5

• Need opposite signs (since $ac$ is negative)
• Larger number is positive (since $b$ is positive)
• Try: 8 and -3 → 8 + (-3) = 5 ✅ and 8 × (-3) = -24 ✅

Step 3: Rewrite the middle term $2x^2 + 5x - 12 = 2x^2 + 8x - 3x - 12$

Step 4: Factor by grouping $= (2x^2 + 8x) + (-3x - 12)$ $= 2x(x + 4) - 3(x + 4)$ $= (2x - 3)(x + 4)$ ✅

Check: $(2x - 3)(x + 4) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12$ ✅

Common Mistakes to Avoid ⚠️

Mistake 1: Forgetting to check for GCF first ❌ Factor $6x^2 + 15x + 6$ directly ✅ Factor out 3 first: $3(2x^2 + 5x + 2)$, then use AC method on what remains

Mistake 2: Wrong signs when finding the two numbers ❌ For $2x^2 - 7x + 3$, using 1 and 6 (adds to 7, not -7) ✅ Use -1 and -6 (both negative since $c$ is positive and $b$ is negative)

Mistake 3: Not matching binomials in Step 4 ❌ Getting $x(2x + 1) + 3(x + 2)$ - these don't match! ✅ Should get $x(2x + 1) + 3(2x + 1)$ - identical binomials

Mistake 4: Incorrect grouping factorization ❌ From $(3x^2 - 6x) + (4x - 8)$, getting $x(3x - 6) + 4(x - 2)$ ✅ Factor completely: $3x(x - 2) + 4(x - 2)$

Pro Tip: Always check your answer by multiplying it back out!

Practice: Complex Trinomials 🎯

Factor each trinomial using the AC Method.

Remember:

1. Check for GCF first!
2. Multiply $a \times c$
3. Find two numbers that multiply to $ac$ and add to $b$
4. Rewrite and group
5. Factor out common binomial

You need to answer 4 questions correctly in a row to proceed to the mini-boss.

Mini-Boss Challenge ⚔️

Face The Trinomial Master!

Part 4 Complete! Outstanding! 🌟

You've conquered the AC Method AND defeated The Trinomial Master!

Remember:

• ✅ The AC Method works for any trinomial
• ✅ Multiply $a \times c$, then find two numbers
• ✅ Factor by grouping - the binomials should match!

Next Up: Part 5 - Special Patterns

In the next part, you'll learn to recognize perfect square trinomials and other special patterns that make factoring even faster!

Ready to learn the shortcuts? Click "Next"!

Part 5: Special Patterns 🎨

Almost there! Let's learn to recognize special trinomials that have shortcuts.

Perfect Square Trinomials

These are special trinomials that factor into a binomial squared!

Why learn these?

• ⚡ They're faster than regular factoring
• ✅ They appear frequently in algebra and calculus
• 🎯 Recognizing them saves time on tests

What You'll Learn:

• 🎨 How to recognize perfect square trinomials
• 📝 The two patterns to memorize
• 🔍 How to verify your answer

Perfect Square Trinomials 🎨

These are special trinomials that factor into a binomial squared!

The Two Patterns: $a^2 + 2ab + b^2 = (a + b)^2$ $a^2 - 2ab + b^2 = (a - b)^2$

How to Recognize Them:

Step 1: Check if the first term is a perfect square

• Is it $x^2$, $4x^2$, $9x^2$, $16x^2$, etc.?

Step 2: Check if the last term is a perfect square

• Is it $1$, $4$, $9$, $16$, $25$, $36$, etc.?

Step 3: Check if the middle term equals $2ab$

• Take the square roots from steps 1 and 2
• Multiply them together and double it
• Does it match the middle term?

Step 4: Write the answer

• If all checks pass: $(\text{first root} \pm \text{second root})^2$
• Use $+$ if middle term is positive, $-$ if negative

Quick Check: Expand your answer to verify!

Example 1: Basic Perfect Square ✨

Factor: $x^2 + 6x + 9$

Step 1: Is $x^2$ a perfect square?

• Yes! $\sqrt{x^2} = x$ ✅

Step 2: Is $9$ a perfect square?

• Yes! $\sqrt{9} = 3$ ✅

Step 3: Is the middle term $2ab$?

• $2 \times x \times 3 = 6x$ ✅
• Matches the middle term!

Step 4: Write the answer

• $x^2 + 6x + 9 = (x + 3)^2$ ✅

Verify: $(x + 3)^2 = x^2 + 6x + 9$ ✅

Example 2: Negative Middle Term 🔄

Factor: $4x^2 - 12x + 9$

Step 1: Is $4x^2$ a perfect square?

• Yes! $\sqrt{4x^2} = 2x$ ✅

Step 2: Is $9$ a perfect square?

• Yes! $\sqrt{9} = 3$ ✅

Step 3: Is the middle term $2ab$?

• $2 \times 2x \times 3 = 12x$
• We have $-12x$ (negative) ✅

Step 4: Write the answer

• Use minus sign: $(2x - 3)^2$ ✅

Verify: $(2x - 3)^2 = 4x^2 - 12x + 9$ ✅

Example 3: Larger Coefficients ⚡

Factor: $25x^2 - 30x + 9$

Step 1: Is $25x^2$ a perfect square?

• Yes! $\sqrt{25x^2} = 5x$ ✅

Step 2: Is $9$ a perfect square?

• Yes! $\sqrt{9} = 3$ ✅

Step 3: Is the middle term $2ab$?

• $2 \times 5x \times 3 = 30x$
• We have $-30x$ (negative) ✅

Step 4: Write the answer

• Use minus sign: $(5x - 3)^2$ ✅

Verify: $(5x - 3)^2 = 25x^2 - 30x + 9$ ✅

Example 4: Two Variables 🌟

Factor: $9x^2 + 24xy + 16y^2$

Step 1: Is $9x^2$ a perfect square?

• Yes! $\sqrt{9x^2} = 3x$ ✅

Step 2: Is $16y^2$ a perfect square?

• Yes! $\sqrt{16y^2} = 4y$ ✅

Step 3: Is the middle term $2ab$?

• $2 \times 3x \times 4y = 24xy$ ✅

Step 4: Write the answer

• $(3x + 4y)^2$ ✅

Verify: $(3x + 4y)^2 = 9x^2 + 24xy + 16y^2$ ✅

Common Mistakes to Avoid ⚠️

Mistake 1: Forgetting to check the middle term ❌ Seeing $x^2 + 5x + 9$ and writing $(x + 3)^2$ ✅ Check: $2 \times x \times 3 = 6x$, not $5x$ → NOT a perfect square, use AC method

Mistake 2: Forgetting to check for GCF first ❌ Factoring $2x^2 + 12x + 18$ as a perfect square ✅ Factor out 2 first: $2(x^2 + 6x + 9) = 2(x + 3)^2$

Mistake 3: Wrong sign in the factored form ❌ For $x^2 - 10x + 25$, writing $(x + 5)^2$ ✅ Middle term is negative, so: $(x - 5)^2$

Mistake 4: Not checking if first/last terms are perfect squares ❌ Trying to use perfect square pattern on $2x^2 + 8x + 8$ ✅ $2x^2$ is NOT a perfect square; factor out 2 first: $2(x^2 + 4x + 4) = 2(x + 2)^2$

Pro Tip: If it's not a perfect square trinomial, no problem! Just use the regular trinomial factoring methods you already know.

Practice: Mixed Factoring 🎯

Factor each expression. Some may be perfect squares, others may need different methods.

Remember the complete strategy:

1. Always check for GCF first!
2. Count the terms (2? difference of squares; 3? trinomial)
3. For trinomials: Is it a perfect square? Check the pattern!
4. If not perfect square, use simple or AC method
5. Always verify your answer!

You need to answer 4 questions correctly in a row to proceed to the mini-boss.

Mini-Boss Challenge ⚔️

Face The Pattern Sage!

Part 5 Complete! You're a Pattern Master! ✨

You've conquered perfect square trinomials AND defeated The Pattern Sage!

Remember:

• ✅ $a^2 + 2ab + b^2 = (a + b)^2$
• ✅ $a^2 - 2ab + b^2 = (a - b)^2$
• ✅ Check: Is the middle term twice the product?

Next Up: Part 6 - Complete Strategy & Mixed Practice

In the final part, you'll learn the complete factoring strategy (what to try when) and practice mixing all the techniques together!

Ready to master factoring? Click "Next"!

Part 6: Complete Strategy & Mastery 🏆

Final part! Let's put it all together.

Your Complete Factoring Toolkit

You've learned all the major techniques. Now it's time to master when to use each one!

What You'll Learn:

• 🎯 The complete factoring strategy (step-by-step)
• 🔄 How to decide which technique to use
• 💪 Mixed practice with all techniques combined

Step 1: Always Check for GCF First! 🔍

This is the most important step - never skip it!

Why start with GCF?

• Makes the remaining expression simpler
• Often reveals hidden patterns
• Required for complete factorization

Examples:

Easy to spot: $6x^2 + 9x = 3x(2x + 3)$

Less obvious: $2x^2 - 50 = 2(x^2 - 25) = 2(x - 5)(x + 5)$ ↑ After factoring GCF, we see a difference of squares!

Must do it: $4x^2 + 12x + 9$

• GCF is 1 (prime coefficients), but check anyway!
• This is a perfect square: $(2x + 3)^2$ ✅

Pro Tip: Even if the GCF is 1, checking takes only seconds and prevents errors!

Step 2: Count the Terms 📊

After factoring out GCF, count what remains:

2 Terms? → Difference of Squares $x^2 - 25 = (x - 5)(x + 5)$ $4x^2 - 9 = (2x - 3)(2x + 3)$

3 Terms? → Trinomial (check in this order):

1️⃣ Is it a perfect square?

• Check if $a^2 \pm 2ab + b^2$
• $x^2 + 6x + 9 = (x + 3)^2$ ✅

2️⃣ Is leading coefficient 1?

• Use simple trinomial method
• $x^2 + 5x + 6 = (x + 2)(x + 3)$ ✅

3️⃣ Leading coefficient ≠ 1?

• Use AC method
• $2x^2 + 7x + 3 = (2x + 1)(x + 3)$ ✅

4+ Terms? → Grouping $x^3 + 2x^2 + 3x + 6 = (x + 2)(x^2 + 3)$

Step 3: Check if You Can Factor Further ✅

Never stop at the first factorization - always check each factor!

Example 1: Hidden difference of squares $3x^2 - 75$

• ❌ WRONG: Stop at $3(x^2 - 25)$
• ✅ RIGHT: $3(x^2 - 25) = 3(x - 5)(x + 5)$

Example 2: Factor can be factored more $2x^3 + 10x^2 + 12x$

• Step 1: Factor GCF → $2x(x^2 + 5x + 6)$
• Step 2: Factor trinomial → $2x(x + 2)(x + 3)$ ✅

Example 3: GCF in factors $6x^2 + 21x + 9$

• Factor GCF first: $3(2x^2 + 7x + 3)$
• Factor trinomial: $3(2x + 1)(x + 3)$ ✅
• If we had: $(6x + 3)(x + 3)$ → NOT fully factored! ($6x + 3$ has GCF of 3)

How to check: Each factor should have no common factors (except 1)

Common Strategy Mistakes ⚠️

Mistake 1: Skipping the GCF check ❌ $2x^2 + 8x + 6 = (2x + 2)(x + 3)$ → NOT fully factored! ✅ Factor out 2 first: $2(x^2 + 4x + 3) = 2(x + 1)(x + 3)$

Mistake 2: Not factoring completely ❌ $x^4 - 16 = (x^2 - 4)(x^2 + 4)$ → NOT done! ✅ $x^2 - 4$ is difference of squares: $(x - 2)(x + 2)(x^2 + 4)$

Mistake 3: Using the wrong technique for the number of terms ❌ Trying to use difference of squares on $x^2 + 25$ (sum of squares doesn't factor!) ✅ Recognize that $x^2 + 25$ is prime over the reals

Mistake 4: Forgetting to check your work

• Always multiply your factors back together
• If it doesn't match the original, find your error

Golden Rule: GCF first, count terms, factor completely, verify!

Mixed Practice: All Techniques! 🎯

Now for the ultimate challenge - mixed practice with all factoring techniques!

You'll need to decide which method to use for each problem. Remember the strategy:

1. ✅ Check for GCF first (always!)
2. ✅ Count the terms (2, 3, or 4+)
3. ✅ Apply the appropriate technique
4. ✅ Check if you can factor further
5. ✅ Verify by expanding

You need to answer 5 questions correctly in a row to proceed to the final boss!

🔥 ULTIMATE BOSS BATTLE 🔥

Face The Factoring Grandmaster - the ultimate test of all your skills!

This is it - the final challenge. Victory requires mastery of every technique!

🎉 CONGRATULATIONS! You've Mastered Factoring! 🎉

You defeated The Factoring Grandmaster!

You now have all the tools you need to factor any polynomial:

✅ Part 1: GCF - Always check first ✅ Part 2: Difference of Squares - $a^2 - b^2$ ✅ Part 3: Simple Trinomials - When $a = 1$ ✅ Part 4: Complex Trinomials - AC Method ✅ Part 5: Special Patterns - Perfect square trinomials ✅ Part 6: Complete Strategy - Know what to try when

Your Factoring Checklist:

1. 🔍 Check for GCF first (always!)
2. 📊 Count the terms
3. 🎯 Apply the right technique
4. ✅ Factor completely
5. 🔄 Verify by multiplying back

Remember:

• Practice makes perfect - the more you factor, the faster you'll recognize patterns
• Always factor completely - check each factor to see if it can be factored further
• Verify your answer by multiplying the factors back together

🎮 Competitive Mode Unlocked! 🎮

Ready to test your speed? You've unlocked Competitive Mode where you can:

• ⚡ Race against the clock
• 🏆 Compete on the leaderboard
• 💪 Practice all factoring types under pressure
• 📊 Track your best times and accuracy

Next Steps:

• Click the competitive mode button to start racing!
• Use factoring to solve quadratic equations
• Apply factoring to simplify rational expressions

You're now a Factoring Master! Keep practicing and you'll be unstoppable! 🚀