Lecture on Factorising Algebraic Expressions

Introduction to Factorising

• Factorising involves breaking down numbers or expressions into their factors.
• Example: 55 can be factorised into 5 x 11 or 1 x 55.
• Algebraic terms can also be factorised, e.g., 3x = 3 x x.

Factorising Expressions

• Involves taking out the highest common factor (HCF) from the expression.
• Example: 3(x + 5)
  • Factorise each term:
    • 3x = 3 x x
    • 15 = 3 x 5
  • HCF is 3
  • Factorised form: 3(x + 5)

Importance of Factorising

• Useful for simplifying expressions, especially in equations and fractions.
• Helps in cancelling terms to simplify algebraic expressions.

Examples

Example 1

• Expression: 6ab - a²b
• Factorisation:
  • Factors of 6ab = 6 x a x b
  • Factors of a²b = a x a x b
  • HCF = ab
  • Factorised form: ab(6 - a)
  • Double-check by expanding back.

Example 2

• Expression: pqr - prt - qs
• Factorisation:
  • Common factor is q
  • Factorised form: q(pr - rt - s)

Example 3

• Expression: 25p² - 10p
• Factorisation:
  • Factors of 25 = 5 x 5, 10 = 5 x 2
  • Common factor = 5p
  • Factorised form: 5p(5p - 2)

Example 4

• Expression: 6p⁴ - 12p
• Factorisation:
  • Factors: p⁴ = p x p x p x p
  • HCF = 6p
  • Factorised form: 6p(p³ - 2)

Summary

• Factorising involves identifying and extracting the highest common factors from expressions.
• Essential for simplifying algebraic expressions.
• Check correctness by expanding back the factorised expression.

Conclusion

• Factorising is a fundamental skill in algebra.
• Essential for simplifying complex algebraic expressions and solving equations.