Overview
This lecture reviews the core concepts from the Grade 9 algebra unit, covering exponent laws, polynomials, collecting like terms, distributive property, and solving equations, including word problems.
Exponent Laws
- A power is written as base^exponent, meaning the base multiplied by itself exponent times (e.g., 5² = 5 × 5).
- Product of powers: xᵃ × xᵇ = x^(a+b) (add exponents for same base).
- Quotient of powers: xᵃ ÷ xᵇ = x^(a-b) (subtract exponents for same base).
- Power of a power: (xᵃ)ᵇ = x^(a×b) (multiply exponents).
- Power of a quotient: (a/b)^x = a^x / b^x (apply exponent to numerator & denominator).
- Power of a product: (ab)^x = a^x × b^x (apply exponent to each factor).
- Anything to the power of zero is 1.
- Negative exponent: x^(-a) = 1 / x^a (write as reciprocal).
Polynomials and Terms
- A term is a product of numbers and variables (e.g., 2x).
- A polynomial is one or more terms linked by + or - (e.g., 4x² + 3x + 1).
- Monomial: 1 term; Binomial: 2 terms; Trinomial: 3 terms; 4+ terms: "n-term polynomial."
- The degree of a term: sum of exponents on all variables in the term.
- The degree of a polynomial: highest degree among its terms.
Collecting Like Terms
- Like terms have identical variables and exponents (coefficients can differ).
- Combine like terms by adding/subtracting their coefficients, keeping variables/exponents unchanged.
- Terms that have different variables or exponents are not like terms.
Distributive Property
- Multiply each term inside brackets by the term outside (a(b + c) = ab + ac).
- Useful for expanding expressions and eliminating brackets.
- When multiplying polynomials and combining like terms, organize by descending degree.
Solving Equations
- To solve an equation, isolate the variable by performing operations to both sides.
- For one-step: use the opposite operation (subtract/add/multiply/divide).
- For two-step: isolate the variable term, then solve for the variable.
- When variables appear on both sides, get all variable terms on one side and constants on the other.
- For equations with fractions, multiply both sides by the common denominator to clear fractions.
- Cross-multiplication can be used when a single fraction equals another single fraction.
- For word problems, define variables, write an equation, and solve.
Key Terms & Definitions
- Exponent — number indicating how many times the base is multiplied by itself.
- Base — the number or variable being multiplied in a power.
- Polynomial — an expression of one or more terms added or subtracted together.
- Degree (of term/polynomial) — sum of exponents in a term; for a polynomial, it's the highest degree of its terms.
- Like Terms — terms with the exact same variables and exponents.
- Distributive Property — multiplying a term by each term inside brackets.
Action Items / Next Steps
- Watch the Linear Relations (Part 2) and Geometry (Part 3) videos for a complete Grade 9 math review.
- Practice problems on exponent laws, polynomials, and equation-solving for exam preparation.