1. What is Factorization?

To factorize means to break down a number or an algebraic expression into a product of simpler parts called factors. For example, 12 can be written as 2 × 2 × 3, and the algebraic expression x² - 9 can be written as (x + 3)(x - 3).

Definition

Factor: A number or algebraic expression that divides another number or expression exactly, without leaving a remainder.
Factorization: The process of expressing a given number or expression as a product of its factors.

Importance of Factorization

• Simplifies complex algebraic expressions
• Makes solving equations easier
• Helps in finding zeros or roots of polynomials
• Used in arithmetic, geometry, and real-world problem solving

2. Factors of Numbers

The process of finding the factors of a whole number is called prime factorization. Every number can be written as a product of prime numbers in only one way (apart from the order of the factors).
Example: 36 = 2 × 2 × 3 × 3 = 2² × 3²

Prime Factorization Method

1. Divide the number by the smallest prime (2, 3, 5, ...)
2. Continue dividing until only 1 remains
3. List all the primes used as factors

Practice: Factorize 210.
Solution: 210 ÷ 2 = 105, 105 ÷ 3 = 35, 35 ÷ 5 = 7, 7 ÷ 7 = 1. So, 210 = 2 × 3 × 5 × 7.

3. Factors of Algebraic Expressions

Just as with numbers, algebraic expressions can be factorized into simpler expressions that multiply to give the original expression. This is especially useful for polynomials.

Example

• 6x²y = 2 × 3 × x × x × y
• x² - 16 = (x + 4)(x - 4)

4. Common Methods of Factorization

A. Taking Out the Common Factor

If all terms in an expression have a common factor, take it outside the bracket.
Example: 4x + 8 = 4(x + 2)

• Factorize 15xy - 20xz = 5x(y - (4z))
• Factorize 18a²b + 27ab² = 9ab(2a + 3b)

B. Factorization by Grouping

When terms can be grouped into pairs with common factors, group and factorize each, then factor the whole.
Example: ab + ac + xb + xc = (ab + ac) + (xb + xc) = a(b + c) + x(b + c) = (a + x)(b + c)

• Factorize: xy + 3y + 2x + 6 = (xy + 3y) + (2x + 6) = y(x + 3) + 2(x + 3) = (y + 2)(x + 3)
• Factorize: a² + ab + 2a + 2b = (a² + ab) + (2a + 2b) = a(a + b) + 2(a + b) = (a + 2)(a + b)

C. Using Algebraic Identities

Recognize patterns that match well-known algebraic identities:

• Difference of squares: a² - b² = (a + b)(a - b)
• Perfect square trinomial: a² + 2ab + b² = (a + b)²
• Sum/difference of cubes: a³ + b³ = (a + b)(a² - ab + b²); a³ - b³ = (a - b)(a² + ab + b²)

• Factorize x² - 9: x² - 3² = (x + 3)(x - 3)
• Factorize y² + 6y + 9: (y + 3)²
• Factorize a³ + 8: a³ + 2³ = (a + 2)(a² - 2a + 4)

D. Factorizing Quadratic Trinomials

For quadratic expressions of the form ax² + bx + c:

1. Find two numbers whose product is a × c and whose sum is b
2. Split the middle term, then group and factorize

Example: x² + 5x + 6
Product = 6, Sum = 5. The numbers are 2 and 3.
x² + 2x + 3x + 6 = x(x + 2) + 3(x + 2) = (x + 3)(x + 2)

• Factorize 2x² + 7x + 3: Product = 6, Sum = 7 → 6 and 1
  2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)
• Factorize x² - 2x - 8: Product = -8, Sum = -2 → 2 and -4
  x² + 2x - 4x - 8 = x(x + 2) - 4(x + 2) = (x - 4)(x + 2)

E. Factorization of Cubic Expressions

Cubic expressions can sometimes be factorized by grouping, using the sum or difference of cubes, or by using the factor theorem for higher algebra.

• Factorize x³ - 8: x³ - 2³ = (x - 2)(x² + 2x + 4)
• Factorize x³ + 3x² - 4x - 12: Group: (x³ + 3x²) + (-4x - 12) = x²(x + 3) - 4(x + 3) = (x + 3)(x² - 4) = (x + 3)(x - 2)(x + 2)

5. Special Types of Factorization

• Factorizing expressions with negative or fractional coefficients
• Expressions with four or more terms (higher polynomials): Use grouping or substitution
• Factorizing by rearranging terms: Always look for possible groups or patterns
• Expressions involving parameters (like a, b, c): Use the same methods, but be careful with calculations

Practice Example

Factorize 3x² - 12x + 12.
First, factor out 3: 3(x² - 4x + 4) = 3(x - 2)²

6. Applications of Factorization

• Simplifying algebraic fractions: Factorize numerator and denominator to cancel common factors.
• Solving quadratic equations: Set the factorized expression equal to zero and solve each factor.
• Geometry: Finding side lengths when area or volume is given.
• Physics: Breaking down formulas for easier calculations.
• Mathematical modeling: Factorization helps find solutions to equations that describe real-world situations.

Real-World Example

The area of a rectangular garden is x² + 7x + 12 square meters. Factorize and find possible length and breadth.
Factorizing: x² + 7x + 12 = (x + 3)(x + 4)
So, possible dimensions are (x + 3) m and (x + 4) m.

7. Common Errors and How to Avoid Them

• Forgetting to take out the highest common factor first.
• Trying to factorize when the expression is already in simplest form.
• Missing signs (positive/negative) during grouping or splitting terms.
• Incorrect splitting of middle terms in trinomials.
• Not checking by multiplying factors to see if they give the original expression.
• Leaving answers with brackets that can be further factorized.

Tip: Always check your final answer by multiplying the factors.

8. Practice Problems

Basic

1. Factorize: 6y + 9
2. Factorize: 2x² + 4x
3. Factorize: x² - 25
4. Factorize: a² + 8a + 16
5. Factorize: x³ - 64

Intermediate

6. Factorize: 5xy + 15y - 2x - 6
7. Factorize: 4a² - 12a + 9
8. Factorize: z³ + 27
9. Factorize: 3m² - 12mn + 12n²
10. Factorize: 2x² - 9x + 10

Advanced

11. Factorize completely: 12x³y - 8x²y² + 4xy³
12. Factorize: x⁴ - 16
13. Factorize: t² - 6t + 9 - 25s²
14. Factorize: 2p³q² - 18pq²
15. Factorize: a³ + b³ + c³ - 3abc

9. Solutions

1. 6y + 9 = 3(2y + 3)
2. 2x² + 4x = 2x(x + 2)
3. x² - 25 = (x + 5)(x - 5)
4. a² + 8a + 16 = (a + 4)²
5. x³ - 64 = (x - 4)(x² + 4x + 16)
6. 5xy + 15y - 2x - 6 = (5xy + 15y) + (-2x - 6) = 5y(x + 3) - 2(x + 3) = (x + 3)(5y - 2)
7. 4a² - 12a + 9 = (2a - 3)²
8. z³ + 27 = (z + 3)(z² - 3z + 9)
9. 3m² - 12mn + 12n² = 3(m² - 4mn + 4n²) = 3(m - 2n)²
10. 2x² - 9x + 10: Product = 20, Sum = -9 → -4 and -5
    2x² - 4x - 5x + 10 = 2x(x - 2) - 5(x - 2) = (x - 2)(2x - 5)
11. 12x³y - 8x²y² + 4xy³ = 4xy(3x² - 2xy + y²)
12. x⁴ - 16 = (x² + 4)(x² - 4) = (x² + 4)(x + 2)(x - 2)
13. t² - 6t + 9 - 25s² = (t - 3)² - (5s)² = (t - 3 + 5s)(t - 3 - 5s)
14. 2p³q² - 18pq² = 2pq²(p² - 9) = 2pq²(p + 3)(p - 3)
15. a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Conclusion

Factorization is a critical skill for simplifying and solving algebraic problems. Whether breaking down complex polynomials or solving practical geometry and arithmetic problems, factorization allows you to see patterns and make calculations easier. By learning and practicing different techniques, and by checking your answers, you will become confident and efficient in your mathematical journey.