Overview
This lecture covers advanced algebra practice questions, focusing on combining skills such as simplifying, multiplying, dividing, and factoring algebraic expressions, as well as evaluating using formulas and substitution.
Simplifying Like Terms
- Combine terms with the same variables and exponents (e.g., 7p² - 4p² = 3p²).
- Only like terms (same variables, same powers) can be added or subtracted.
- Final simplified form: 3p² - 12p + 1.
Multiplying and Dividing Algebraic Terms
- When multiplying, multiply coefficients and add exponents for matching bases (e.g., 6a² × 2a³ = 12a⁵).
- Division: Divide coefficients and subtract exponents when bases match (e.g., 15v⁶ ÷ 3v² = 5v⁴).
- Anything divided by itself is 1 (e.g., x/x = 1).
Operations with Fractions and Algebraic Expressions
- For multiplying fractions: multiply numerators and denominators, then simplify by common factors (e.g., (2x/3) × (4/x) = 8/3).
- For dividing fractions: multiply by the reciprocal (e.g., (2c/5) ÷ (3d/4) = (2c/5) × (4/3d) = 8c/15d).
- Simplify within fractions when possible before multiplying.
Substitution and Evaluation
- Substitute given values into algebraic expressions and follow order of operations (e.g., substitute a=4, b=–2 into 3a + b).
- Use brackets to clarify the numerator/denominator before simplifying the result.
- Express negative fractions with the negative sign in the numerator.
Using Formulas
- Substitute known values into formulas (e.g., converting °F to °C: C = 5/9(F – 32)).
- Calculate step-by-step, simplifying where possible (e.g., C = 25°C when F = 77).
Expanding and Factorizing
- To expand, multiply each term inside brackets by the term outside (e.g., 3(x + y – z) = 3x + 3y – 3z).
- To factorize, find the highest common factor in coefficients and variables, then factor it out (e.g., 3x² – 12x = 3x(x – 4)).
- Factor out common factors from all terms.
Applying Order of Operations (BIDMAS/BODMAS)
- Follow brackets, indices, division/multiplication, addition/subtraction rules in complex expressions.
- Simplify terms in brackets first before applying multiplication or other operations.
Key Terms & Definitions
- Like Terms — Terms with the same variable(s) and exponent(s).
- Coefficient — The numerical part of a term that multiplies the variables.
- Exponent — The power to which a variable is raised.
- Reciprocal — The flipped form of a fraction (e.g., reciprocal of 3/4 is 4/3).
- Factorize — Rewrite an expression as a product of factors.
Action Items / Next Steps
- Review practice questions on simplifying, multiplying, dividing, expanding, and factorizing algebraic expressions.
- Complete worksheet problems similar to those discussed, especially on fraction operations and substitution.
- Practice evaluating expressions and using formulas with substitution for mastery.