Lecture Notes: Expanding Binomials and Algebraic Expressions

Overview

This lecture focuses on algebraic expressions, specifically expanding products of binomials and simplifying expressions involving brackets. It covers several levels of complexity, providing examples and exercises for each.

Levels of Complexity

1. Level 1: Collecting like terms when 1 term is repeated.
2. Level 2: Collecting like terms when 2 terms are repeated.
3. Level 3: Multiplying a single positive integer over a bracket.
4. Level 4: Multiplying a single negative integer over a bracket.
5. Level 5: Multiplying a variable over a bracket.
6. Level 6: Expanding products of two simple binomials.
7. Level 7: Expanding products of two binomials.
8. Level 8: Squaring a binomial.
9. Level 9: Simplifying complex expressions with brackets.
10. Level 10: Expanding products of three binomials (cubic expressions).

Level 6: Expanding Products of Two Simple Binomials

• Objective: Simplify expressions by removing brackets.
• Instructions:
  • Order terms: For questions 1-6, follow alphabetical order (F.O.I.L. order).
  • Order terms: For questions 7-12, order variables in decreasing power.

Example Problems

1. (a+b)(c+d)
2. (b+c)(de)
3. (cd)(e+f)
4. (de)(fg)
5. (e5)(f2 g)
6. (2 f7)(3 g+10)
7. (g+9)(g7)
8. (h10)(h+8)
9. (i11)(i+9)
10. (j+17)(j10)
11. (k+12)(k11)
12. (m+11)(m12)

Example Solutions

• Level 1 Example: 5d - (2d + 2) simplifies to 3d - 2.
• Level 2 Example: (9d - 7) - (5d - 2) simplifies to 4d - 5.
• Level 3 Example: 4(2d + 4) becomes 8d + 16.
• Level 4 Example: -5(5d + 4) becomes -25d - 20.
• Level 5 Example: 4d(5d - 5) simplifies to 20d^2 - 20d.
• Level 6 Example: (d + 7)(d - 2) simplifies to d^2 + 5d - 14 using F.O.I.L.

Techniques and Tips

• F.O.I.L. Method: First, Outer, Inner, Last for multiplying binomials.
• Mathematical Order: Type answers in decreasing powers of the variable for consistency.

Related Resources

• Exam Style Questions: Practice with GCSE or IB/A-level style questions.
• Binomial Theorem Exercises: Focus on expanding powers of binomial expressions.
• More Algebra Resources

Additional Information

• MathQuill: Utilized for typing mathematical notation online.
• Transum Subscription: Access answers and additional resources by logging into your Transum account. A subscription is available for teachers, tutors, or parents.

Note

Answers are verified by matching typed responses with predefined correct answers, emphasizing the importance of correct order and format.