Can You REALLY Factorise Like a PRO in Just 5 Days?

Advanced factorization often requires combining multiple techniques to solve complex expressions. At HappyMath, we understand that developing this skill requires both practice and a systematic approach. In this guide, we'll explore how to tackle expressions that require a combination of:

1. Finding common factors
2. Using the difference of squares formula
3. Factoring by grouping
4. Simplifying surds (square roots) within expressions

The key to success lies in recognizing which techniques to apply and in what order. Let's explore the systematic approach to solving these challenging problems.

Can You REALLY Factorise Like a PRO in Just 5 Days?

Factorizing Expressions with Surds and Fractions

When dealing with expressions containing surds (square roots) in fraction form, we apply the difference of squares formula while carefully handling the fractional parts.

Examples:

Example 1: x² - (√2/3)²

Step 1: Identify this as a difference of squares pattern.

• x² - (√2/3)² = (x - √2/3)(x + √2/3)

Example 2: x² - (√3/2)²

Step 1: Apply the difference of squares formula.

• x² - (√3/2)² = (x - √3/2)(x + √3/2)

Example 3: x² - (√7/4)²

Step 1: Apply the difference of squares formula.

• x² - (√7/4)² = (x - √7/4)(x + √7/4)

Example 4: x² - (√5/6)²

Step 1: Apply the difference of squares formula.

• x² - (√5/6)² = (x - √5/6)(x + √5/6)

Factorizing Expressions with Binomials and Surds

For expressions involving binomials and surds, we often need to restructure the expression to identify the difference of squares pattern.

Examples:

Example 5: (x - 2)² - 20

Step 1: Recognize this as (x - 2)² - (√20)² Step 2: Factor using the difference of squares formula:

• (x - 2)² - (√20)² = [(x - 2) - √20][(x - 2) + √20]
• = (x - 2 - √20)(x - 2 + √20)

Step 3: Simplify the surd if possible:

• √20 = √(4 × 5) = 2√5
• = (x - 2 - 2√5)(x - 2 + 2√5)

Example 6: (x + 4)² - 27

Step 1: Recognize this as (x + 4)² - (√27)² Step 2: Factor using the difference of squares formula:

• (x + 4)² - (√27)² = [(x + 4) - √27][(x + 4) + √27]
• = (x + 4 - √27)(x + 4 + √27)

Step 3: Simplify the surd:

• √27 = √(9 × 3) = 3√3
• = (x + 4 - 3√3)(x + 4 + 3√3)

Example 7: (x + 1)² - 75

Step 1: Recognize this as (x + 1)² - (√75)² Step 2: Factor using the difference of squares formula:

• (x + 1)² - (√75)² = [(x + 1) - √75][(x + 1) + √75]
• = (x + 1 - √75)(x + 1 + √75)

Step 3: Simplify the surd:

• √75 = √(25 × 3) = 5√3
• = (x + 1 - 5√3)(x + 1 + 5√3)

Example 8: (x - 7)² - 40

Step 1: Recognize this as (x - 7)² - (√40)² Step 2: Factor using the difference of squares formula:

• (x - 7)² - (√40)² = [(x - 7) - √40][(x - 7) + √40]
• = (x - 7 - √40)(x - 7 + √40)

Step 3: Simplify the surd:

• √40 = √(4 × 10) = 2√10
• = (x - 7 - 2√10)(x - 7 + 2√10)

Factorizing Expressions with Variables in Surds

Some advanced expressions include variables inside the square root. These require careful application of the difference of squares formula.

Examples:

Example A: (√3·x)² - 4

Step 1: Factor using the difference of squares formula:

• (√3·x)² - 4 = [(√3·x) - 2][(√3·x) + 2]
• = (√3·x - 2)(√3·x + 2)

Example B: (√5·x)² - 9

Step 1: Factor using the difference of squares formula:

• (√5·x)² - 9 = [(√5·x) - 3][(√5·x) + 3]
• = (√5·x - 3)(√5·x + 3)

Example C: (√7·x)² - 5

Step 1: Factor using the difference of squares formula:

• (√7·x)² - 5 = [(√7·x) - √5][(√7·x) + √5]
• = (√7·x - √5)(√7·x + √5)

Factorizing Using Common Factors Before Applying Other Techniques

Often, we need to factor out common terms before applying the difference of squares formula.

Examples:

Example D: 5x² - 120

Step 1: Factor out the common factor 5:

• 5(x² - 24)

Step 2: Apply the difference of squares formula to (x² - 24):

• 5(x² - 24) = 5[(x - √24)(x + √24)]

Step 3: Simplify the surd:

• √24 = √(4 × 6) = 2√6
• = 5[(x - 2√6)(x + 2√6)]
• = 5(x - 2√6)(x + 2√6)

Example E: 3x² - 162

Step 1: Factor out the common factor 3:

• 3(x² - 54)

Step 2: Apply the difference of squares formula to (x² - 54):

• 3(x² - 54) = 3[(x - √54)(x + √54)]

Step 3: Simplify the surd:

• √54 = √(9 × 6) = 3√6
• = 3[(x - 3√6)(x + 3√6)]
• = 3(x - 3√6)(x + 3√6)

Example F: 7x² - 126

Step 1: Factor out the common factor 7:

• 7(x² - 18)

Step 2: Apply the difference of squares formula to (x² - 18):

• 7(x² - 18) = 7[(x - √18)(x + √18)]

Step 3: Simplify the surd:

• √18 = √(9 × 2) = 3√2
• = 7[(x - 3√2)(x + 3√2)]
• = 7(x - 3√2)(x + 3√2)

Example G: 2x² - 96

Step 1: Factor out the common factor 2:

• 2(x² - 48)

Step 2: Apply the difference of squares formula to (x² - 48):

• 2(x² - 48) = 2[(x - √48)(x + √48)]

Step 3: Simplify the surd:

• √48 = √(16 × 3) = 4√3
• = 2[(x - 4√3)(x + 4√3)]
• = 2(x - 4√3)(x + 4√3)

Example H: 5(x + 6)² - 90

Step 1: Distribute the 5:

• 5(x + 6)² - 90 = 5(x + 6)² - 90

Step 2: Factor out 5 from the entire expression:

• 5[(x + 6)² - 18]

Step 3: Apply the difference of squares formula:

• 5[(x + 6)² - 18] = 5[(x + 6) - √18][(x + 6) + √18]

Step 4: Simplify the surd:

• √18 = √(9 × 2) = 3√2
• = 5[(x + 6 - 3√2)(x + 6 + 3√2)]
• = 5(x + 6 - 3√2)(x + 6 + 3√2)

Factorizing by Grouping in Advanced Expressions

Some expressions require rearrangement and factoring by grouping before other techniques can be applied.

Examples:

Example I: 3x - 6y + xy - 2y²

Step 1: Rearrange to group similar terms:

• (3x - 6y) + (xy - 2y²)

Step 2: Factor out common terms from each group:

• 3(x - 2y) + y(x - 2y)

Step 3: Factor out the common binomial:

• (x - 2y)(3 + y)

Example J: ax + 3a - 12 - 4x

Step 1: Rearrange to group similar terms:

• (ax + 3a) + (-12 - 4x)

Step 2: Factor out common terms from each group:

• a(x + 3) - 4(3 + x)

Step 3: Rearrange for a common binomial:

• a(x + 3) - 4(x + 3)

Step 4: Factor out the common binomial:

• (x + 3)(a - 4)

Example K: ax + 5x - 10 - 2a

Step 1: Rearrange to group similar terms:

• (ax + 5x) + (-10 - 2a)

Step 2: Factor out common terms from each group:

• x(a + 5) - 2(5 + a/2)

Step 3: Recognize that (5 + a/2) can be written as (a + 5)/2:

• x(a + 5) - 2(a + 5)/2

Step 4: Simplify:

• x(a + 5) - (a + 5)

Step 5: Factor out the common binomial:

• (a + 5)(x - 1)

Example L: xy - 3y + 12 - 4x

Step 1: Rearrange to group similar terms:

• (xy - 4x) + (-3y + 12)

Step 2: Factor out common terms from each group:

• x(y - 4) - 3(y - 4)

Step 3: Factor out the common binomial:

• (y - 4)(x - 3)

Key Strategies for Advanced Factorization

When tackling complex factorization problems, remember these key strategies:

1. Look for common factors first: Always check if terms share common factors that can be factored out.
   
    
2. Identify standard patterns: Be on the lookout for difference of squares, perfect square trinomials, and other standard patterns.
   
    
3. Simplify surds when possible: Convert expressions like √20 to 2√5 to make the factorization cleaner.
   
    
4. Rearrange strategically: Sometimes rearranging terms can reveal hidden patterns or common factors.
   
    
5. Combine techniques: Many complex expressions require multiple factorization techniques applied in sequence.
   
    
6. Verify your answer: Always check your factorization by multiplying the factors to ensure you get the original expression.
   
    

Common Mistakes to Avoid

When factorizing advanced expressions, watch out for these common pitfalls:

1. Forgetting to simplify surds: Always check if square roots can be simplified.
   
    
2. Missing common factors: Always look for the greatest common factor before applying other techniques.
   
    
3. Incorrect application of the difference of squares: Remember the pattern a² - b² = (a - b)(a + b).
   
    
4. Errors in distributing negative signs: Be careful when factoring out negative terms.
   
    
5. Incomplete factorization: Always check if your answer can be factored further.
   
    

Building Your Advanced Factorization Skills

At HappyMath, we believe that mastering advanced factorization comes through systematic practice and understanding. Start with simpler examples and gradually tackle more complex ones. With time, you'll develop the mathematical intuition to recognize which techniques to apply and in what order.

Remember that factorization is both a science and an art—while there are systematic approaches, developing mathematical intuition through practice is equally important. Keep honing your skills, and you'll soon be able to factorize even the most challenging expressions with confidence and precision.