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WBB • Class 7 • Mathematics (গণিত প্রভা) • Ch 19

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Factorization

Welcome to Chapter 19 "Factorization" of the West Bengal Board of Secondary Education (WBBSE) Class 7 Mathematics (Ganit Prabha) curriculum. This comprehensive guide covers irreducible algebraic factors, greatest common monomial and polynomial extraction, the method of grouping terms, difference of two squares identities ($a^2 - b^2 = (a+b)(a-b)$), perfect square trinomial representations ($a^2 \pm 2ab + b^2 = (a \pm b)^2$), completing the square for higher-order polynomials, middle-term split factoring, and determining the HCF and LCM of algebraic expressions using TargetExams Gold-Standard 5-step pedagogy.

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🔐 Digital Cryptography & RSA Security: How Does Factoring Guard Global Banking?

Multiplying two prime numbers like $17 \times 23 = 391$ takes only a couple of seconds. But if someone gives you only $391$ and asks which two primes produced it, how long would it take to reverse the operation?

Deconstructing a number or algebraic expression into its irreducible prime factors is called Factorization. The entire modern cybersecurity industry—including RSA encryption safeguarding credit cards and passwords—is anchored in the computational difficulty of factorization!

In algebra, factorization is the ultimate tool for decoding polynomial structures, simplifying rational fractions, and solving equations.

Why This Chapter Matters

Before You Begin (Prerequisites)

Concept of algebraic terms, coefficients, like and unlike terms
Polynomial multiplication and division rules
Standard square identities: $(a+b)^2, (a-b)^2$, and $a^2 - b^2$
Arithmetic prime factorization, HCF, and LCM

What You Will Learn (Core Objectives)

Extract the greatest common factor (monomial or binomial) from any expression
Factorize 4-term expressions using the grouping of terms method
Apply the difference of two squares identity ($a^2 - b^2$) repeatedly to complete factorization
Factorize trinomials into perfect squares or by completing the square
Determine the HCF and LCM of algebraic polynomials by prime factorization

Chapter Roadmap & Progression

1 Concept 1: Definition of Factorizat...

2 Concept 2: Grouping Method (Rearran...

3 Concept 3: The Difference of Two Sq...

4 Concept 4: Perfect Square Trinomial...

5 Concept 5: Middle-Term Splitting &...

Complete Concept Guide (100% Curriculum Coverage)

Concept 1: Definition of Factorization & Common Factor Extraction

Step 1

Fundamental Definition

Factorization: Expressing an algebraic sum/difference as the product of two or more irreducible polynomials.

Step 2

Extracting the GCF

Find the highest common numerical factor and the lowest power of each shared variable:
$$12x^2y + 18xy^2 = 6xy(2x + 3y)$$

Step 3

Worked Example

$7a^2b - 14ab = 7ab(a - 2)$, where $7ab$ and $(a-2)$ are irreducible factors.

Step 4

Examiner Trap

When an entire term is factored out, a 1 remains, never 0: $5x + 5 = 5(x + 1)$.

Step 5

Computer Science

Compiler Optimization: Compilers factor common sub-expressions to minimize machine clock cycles.

Concept 2: Grouping Method (Rearranging Polynomial Terms)

Step 1

When to Group

When an expression has 4 or more terms with no overall single common factor across all terms, group them into pairs.

Step 2

Algebraic Structure

$$ax + bx + ay + by = x(a+b) + y(a+b) = (a+b)(x+y)$$

Step 3

Sign Handling Example

$xy - 2y - 3x + 6 = y(x - 2) - 3(x - 2) = (x - 2)(y - 3)$.

Step 4

Caution

Both bracketed binomials must match identically before final factoring.

Step 5

Electronics

Boolean Logic: Grouping reduces logic gate counts in microprocessor circuitry.

Concept 3: The Difference of Two Squares Identity ($a^2 - b^2$)

Step 1

Master Identity

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

Step 2

Cascade Factoring

$$x^4 - y^4 = (x^2 + y^2)(x^2 - y^2) = (x^2 + y^2)(x + y)(x - y)$$

Step 3

Compound Expression

$(2x + 3y)^2 - (x - 2y)^2 = [(2x+3y)+(x-2y)][(2x+3y)-(x-2y)] = (3x+y)(x+5y)$.

Step 4

Trap

Distribute the negative sign correctly when subtracting: $-(x - 2y) = -x + 2y$.

Step 5

Mental Arithmetic

Speed Math: $105^2 - 95^2 = (105+95)(105-95) = 200 \times 10 = 2000$.

Concept 4: Perfect Square Trinomials & Completing the Square

Step 1

Perfect Squares

$$a^2 \pm 2ab + b^2 = (a \pm b)^2$$

Step 2

Quartic Technique

$$x^4 + x^2y^2 + y^4 = (x^2 + y^2)^2 - (xy)^2 = (x^2 + xy + y^2)(x^2 - xy + y^2)$$

Step 3

Worked Example

$a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2)$.

Step 4

Caution

$a^2 + b^2$ cannot be factored over real numbers.

Step 5

Quadratic Equations

Higher Algebra: Completing the square is the foundation of the quadratic formula.

Concept 5: Middle-Term Splitting & Polynomial HCF / LCM

Step 1

Middle-Term Split

$$x^2 + (p + q)x + pq = (x + p)(x + q)$$ e.g. $x^2 + 5x + 6 = (x + 2)(x + 3)$.

Step 2

HCF & LCM Rules

• HCF: Product of common factors only.• LCM: Product of all factors raised to their highest powers.

Step 3

Worked Example

HCF of $x^2 - 4 = (x+2)(x-2)$ and $x^2 - 5x + 6 = (x-2)(x-3)$ is $(x - 2)$.

Step 4

Caution

Never compute HCF/LCM without factorizing completely first.

Step 5

Secondary School Math

Class 8-10 Bridge: Crucial for simplifying algebraic fractions and solving systems of equations.

Key Formulas, Identities & Theorems

Difference of Two Squares Identity

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

The difference of two squared terms equals their sum times their difference.

Perfect Square Trinomials

$$a^2 \pm 2ab + b^2 = (a \pm b)^2 = (a \pm b)(a \pm b)$$

Expressing a 3-term quadratic polynomial as twin linear factors.

Grouping of Terms Identity

$$ax + bx + ay + by = (a+b)(x+y)$$

Pairing terms strategically to extract binomial common factors.

Middle-Term Split Factoring

$$x^2 + (p+q)x + pq = (x + p)(x + q)$$

Where sum of factors equals middle coefficient and product equals constant.

Quartic Polynomial Decomposition

$$a^4 + a^2b^2 + b^4 = (a^2 + ab + b^2)(a^2 - ab + b^2)$$

Completing the square via $(a^2+b^2)^2 - (ab)^2$.

Algebraic HCF & LCM Product Rule

$$\text{Product of Expressions} = \text{HCF} \times \text{LCM}$$

Fundamental relationship between two polynomial expressions and their HCF/LCM.

Conceptual Solved Examples & Case Studies

Example 1

Factorize completely: $x^4 - 81$.

Step-by-Step Solution:

$x^4 - 81$
$= (x^2)^2 - (9)^2$
$= (x^2 + 9)(x^2 - 9)$ [using $a^2 - b^2 = (a+b)(a-b)$]
Now decompose the second factor further:
$= (x^2 + 9)[(x)^2 - (3)^2]$
$= (x^2 + 9)(x + 3)(x - 3)$.
(Note: $(x^2 + 9)$ cannot be factored further over real numbers).

Example 2

Factorize by grouping: $ax - ay - bx + by$.

Step-by-Step Solution:

$ax - ay - bx + by$
$= a(x - y) - b(x - y)$ [Watch signs: factoring out $-b$ changes $+by$ to $-y$]
$= (x - y)(a - b)$.

Example 3

Factorize: $p^2 - 2pq + q^2 - r^2$.

Step-by-Step Solution:

Grouping the first three terms:
$(p^2 - 2pq + q^2) - r^2$
$= (p - q)^2 - r^2$
Applying $a^2 - b^2 = (a+b)(a-b)$:
$= (p - q + r)(p - q - r)$.

Common Misconceptions & Examiner Traps

Common Misconception

Stopping at $x^4 - 16 = (x^2+4)(x^2-4)$ without factoring $(x^2-4)$ further.

Scientific Reality & Correction

Always check if any resulting binomial is still a difference of squares: $(x^2-4) = (x+2)(x-2)$.

Common Misconception

Failing to reverse signs when factoring out a negative common term.

Scientific Reality & Correction

Factoring out $-b$ in $-bx + by$ gives $-b(x - y)$, not $-b(x + y)$.

Common Misconception

Confusing $(a - b)^2$ with $a^2 - b^2$.

Scientific Reality & Correction

$(a - b)^2 = a^2 - 2ab + b^2$ is a perfect square trinomial, whereas $a^2 - b^2 = (a+b)(a-b)$ is a difference of squares.

Geometric Proof & Area Model of Factorization

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All Class 7 Mathematics (গণিত প্রভা) Chapters

Ch 1: Revision of Previous Lessons Ch 2: Ratio Ch 3: Proportion Ch 4: Addition, Subtraction, Multiplication and Division of Integers Ch 5: Concept of Indices Ch 6: Algebraic Operations Ch 7: Construction of Specific Angles with Compass Ch 8: Construction of Triangles Ch 9: Concept of Congruence Ch 10: Approximation Ch 11: Square Root of Fractions Ch 12: Algebraic Formulae Ch 13: Parallel Lines and Transversals Ch 14: Properties of Triangles Ch 15: Time and Distance Ch 16: Double Bar Graph Ch 17: Area of Rectangle and Square Ch 18: Symmetry Ch 19: Factorization Ch 20: Classification of Quadrilaterals Ch 21: Construction of Quadrilaterals Ch 22: Formation and Solution of Equations

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