A common factor is a divisor (a number, a variable, or even an entire bracketed expression) that is shared by every term of a polynomial. Extracting the common factor is the simplest form of factorization and should always be considered first.
The Three-Step Extraction Procedure:
Look at the coefficients: Find the GCD (Greatest Common Divisor) of all numerical coefficients.
Look at the variables: Take each variable that appears in every term, using the lowest power that occurs.
Look at the brackets: If an entire bracketed expression appears as a factor in every term, the whole bracket can be factored out.
Example 1.1.1: Factorize 6x³y² − 9x²y + 3xy
①Numerical GCD: GCD(6, 9, 3) = 3
②Variables: Every term contains x and y; the lowest powers are x¹ and y¹, so the variable part is xy.
③Extract:3xy(2x²y − 3x + 1)
Example 1.1.2: Factorize 5a(b + c) − 3(b + c)
①Identify the bracket: Both terms contain the bracket (b + c).
②Extract:(b + c)(5a − 3)
⚠️ Common Mistake: After extracting, students often forget to check whether the terms inside the new bracket still share a common factor. Always verify that the factorization is complete by multiplying back out and confirming no further extraction is possible.
1.2 Cross Method (AC Method)
📌 The Standard Quadratic Trinomial
ax² + bx + c (a ≠ 0)
The Cross Method (also called the AC Method in Western textbooks) is the standard technique for factorizing a quadratic trinomial where a ≠ 1 or where the factors are not immediately obvious.
� The Cross-Multiplication Principle
Cross-Multiplication diagram: the cross products (2 × 3 and 1 × 1) sum to the middle coefficient b = 7.
The Four-Step Procedure:
Split a into two factors: a = p × q
Split c into two factors: c = r × s
Check the cross-sum: does p × s + q × r = b hold?
Write the answer: (px + r)(qx + s)
Example 1.2.1: Factorize x² + 5x + 6
①a = 1, c = 6, b = 5
②Find two numbers whose sum is 5 and product is 6: 2 and 3.
③Result: (x + 2)(x + 3)
Example 1.2.2: Factorize 2x² + 7x + 3
①a = 2, c = 3, b = 7
②Find two numbers with sum = 7 and product = 2 × 3 = 6: 1 and 6.
💡 Mnemonic: "Two numbers whose sum is the middle and whose product is the ends" — i.e. sum = b and product = a × c.
⚠️ Technique for Leading-Negative Coefficients: If the whole trinomial is preceded by a minus sign, factor the minus out first, then proceed. For example: −2x² + 7x − 3 = −(2x² − 7x + 3) = −(2x − 1)(x − 3)
1.3 Perfect Square Trinomials
📐 The Two Identities
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
🔍 Recognition Pattern
When you see a trinomial, check whether:
The first and last terms are both perfect squares (of some quantity a and some quantity b).
The middle term equals ±2ab (twice the product of the square roots of the first and last terms).
If both conditions hold, the trinomial is a perfect square.
Example 1.3.1: Factorize x² + 6x + 9
①x² = (x)², 9 = (3)²
②Middle term: 6x = 2 × x × 3 ✓
③Sign of middle term is positive → (x + 3)²
Example 1.3.2: Factorize 4x² − 12xy + 9y²
①4x² = (2x)², 9y² = (3y)²
②Middle term: −12xy = −2 × 2x × 3y ✓
③Sign is negative → (2x − 3y)²
� Quick Test: "First square + Last square, and Middle = ±2 × (First root) × (Last root)" — always a perfect square.
1.4 Difference of Squares
📐 The Identity
a² − b² = (a + b)(a − b)
🔍 Recognition Pattern
When you see a binomial in which:
There are only two terms, connected by a minus sign.
⚠️ Important Note: The sum of squaresa² + b²cannot be factorized over the real numbers. It is irreducible in ℝ.
1.5 Completing the Square
📌 When Do We Use It?
When all other methods (common factor, cross method, perfect square, difference of squares, grouping) have been tried and the polynomial still resists factorization, completing the square can be used as a last-resort technique to force the expression into a difference-of-squares form.
🔍 The Procedure
To complete the square on ax² + bx + c:
Compute (b/2)². Insert it and its negation: rewrite the constant term as (b/2)² − (b/2)² + c.
Group the first three terms into a perfect square: (x + b/2)².
Simplify the remaining constant.
Apply the difference-of-squares formula to finish.
Example 1.5.1: Factorize x² + 4x + 1 (a classic DSE question!)
③Rewrite 3 as (√3)²; the expression becomes (x + 2)² − (√3)² — a difference of squares.
④Result: (x + 2 + √3)(x + 2 − √3)
Example 1.5.2: Factorize 2x² + 8x + 6
①First extract the common factor 2: 2(x² + 4x + 3).
②Apply the cross method to the bracket: x² + 4x + 3 = (x + 1)(x + 3).
③Result: 2(x + 1)(x + 3)
📌 Note: Completing the square is generally the last-resort technique. Always try cross method, perfect square, and difference of squares first — they are usually faster and cleaner.
1.6 Factorization by Grouping
📌 The Five-Step Procedure
Count the terms: only consider grouping when there are 4 or more terms.
Split into pairs: group the first two terms together, and the last two terms together.
Extract a common factor from each pair.
Check whether the two pairs now share a common bracket.
Factor out the shared bracket, leaving the remaining brackets multiplied.
②Attempt grouping B: (2x² + 3xy) + (−2y² − 5yz) → x(2x + 3y) − y(2y + 5z) — still no common bracket.
③Multi-variable problems like this often require treating 2x² + 3xy − 2y² as one group and factorizing it with the cross method:
2x² + 3xy − 2y² = (2x − y)(x + 2y) Then factor −y from the remaining piece: −y(2x − y) + ... = ...
⚠️ Important: The grouping is not unique. If one grouping fails to produce a common bracket, try a different pairing. The correct grouping is the one that allows the factorization to continue.
1.7 Comprehensive (Mixed) Problems
Comprehensive problems combine two or more of the techniques above. They usually require a creative first step — pairing, inserting terms, or splitting cleverly — before a familiar identity becomes visible.
📌 Technique 1: Strategic Pairing
Example 1.7.1: Factorize (x + 1)(x + 2)(x + 3)(x + 4) − 24 ("the king of mixed problems")
💡 Strategy for Comprehensive Problems: First observe the structure → then pair or insert terms to expose a familiar identity (perfect square or difference of squares) → finally apply the identity.
1.8 Five Decision Mnemonics & Quick Reference
🧭 The Five-Step Decision Ladder
When you face a polynomial and must decide which technique to apply, walk down this ladder in order:
The Decision Ladder — work top-down:
Is there a common factor? → Extract it. (Always try this first.)
Is it a trinomial? → Use the Cross Method (find two numbers whose sum is b and product is ac).
Is the middle term ±2 × (first root) × (last root)? → Perfect Square.
Is it a binomial of two perfect squares with a minus sign? → Difference of Squares.
Does it have 4 or more terms? → Factorization by Grouping.
None of the above worked? → Completing the Square (insert and subtract).
📊 Quick Reference: The 7 Major Pattern Types
#
Pattern Type
Recognition Clue
Key Step
1
Common Factor Extraction
Every term shares a numerical or variable factor (or bracket).
Find the GCD of the coefficients; choose the lowest power of each common variable.
Two perfect squares with a minus? Difference of Squares.
4+ terms? Grouping — try different pairings until a shared bracket appears.
All else fails? Completing the Square.
⚠️ Five Common Errors to Avoid
Forgetting to verify that the bracketed part after extraction has no remaining common factor.
Mis-assigning the cross-method factors when the constant c is negative (remember to allow negative pair candidates).
Trying to factor a sum of squaresa² + b² — it is irreducible over ℝ.
Stopping at the first failed grouping — the grouping is not unique; try another pairing.
Reaching for completing the square before exhausting cross method, perfect square, and difference of squares.
✏️ Quick Quiz (10 Practice Questions)
Q1. Factorize 12x²y − 8xy²
📌 Click to reveal the answer
Common factor: 4xy.
Answer: 4xy(3x − 2y)
Q2. Factorize x² + 7x + 12
📌 Click to reveal the answer
Find two numbers with sum = 7 and product = 12: 3 and 4.
Answer: (x + 3)(x + 4)
Q3. Factorize x² − 8x + 16
📌 Click to reveal the answer
x² = (x)², 16 = (4)², and 2 × x × 4 = 8x — perfect square (negative sign in the middle).
Answer: (x − 4)²
Q4. Factorize 9x² − 4y²
📌 Click to reveal the answer
9x² = (3x)², 4y² = (2y)² — difference of squares.
Answer: (3x + 2y)(3x − 2y)
Q5. Factorize 2x² + 5x − 3
📌 Click to reveal the answer
Find two numbers with sum = 5 and product = 2 × (−3) = −6. The pair is 6 and −1.
Distribution that yields 5: 2 × (−1) + 1 × 6 = 4; other trial: 2 × 6 + 1 × (−1) = 11. The successful distribution is the one that places 3 next to 2x: 2 × 3 + 1 × (−1) = 5.
Answer: (2x − 1)(x + 3)
Q6. Factorize ax − 3bx + ay − 3by
📌 Click to reveal the answer
Group: (ax − 3bx) + (ay − 3by).
Extract: x(a − 3b) + y(a − 3b). Common bracket (a − 3b).
Answer: (a − 3b)(x + y)
Q7. Factorize x² + 6x + 5
📌 Click to reveal the answer
Find two numbers with sum = 6 and product = 5: 1 and 5.
Answer: (x + 1)(x + 5)
Q8. Factorize 25x² − 1
📌 Click to reveal the answer
25x² = (5x)², 1 = 1² — difference of squares.
Answer: (5x + 1)(5x − 1)
Q9. Factorize x² + 2x − 3
📌 Click to reveal the answer
Find two numbers with sum = 2 and product = −3: −1 and 3.
Answer: (x − 1)(x + 3)
Q10. Factorize (x + y)² − (x − y)²
📌 Click to reveal the answer
Both terms are perfect squares and they are being subtracted. Apply a² − b² = (a + b)(a − b) with a = x + y and b = x − y.
a + b = (x + y) + (x − y) = 2x; a − b = (x + y) − (x − y) = 2y.