Overview
This lecture reviews all core Grade 9 Algebra concepts, including exponent laws, polynomials, collecting like terms, distributive property, and solving equations, with examples for each.
Exponent Laws
- A power is a base number raised to an exponent; the exponent shows how many times the base is multiplied.
- Product of powers: multiply same-base powers by adding exponents, e.g., ( x^a \times x^b = x^{a+b} ).
- Quotient of powers: divide same-base powers by subtracting exponents, e.g., ( x^a / x^b = x^{a-b} ).
- Power of a power: raise a power to another exponent by multiplying exponents, e.g., ( (x^a)^b = x^{ab} ).
- Power of a quotient: exponent applies to numerator and denominator, ( (a/b)^x = a^x / b^x ).
- Power of a product: exponent applies to each factor, ( (ab)^x = a^x b^x ).
- Any base to the power of 0 equals 1, ( x^0 = 1 ).
- Negative exponent: write as reciprocal, ( x^{-a} = 1 / x^a ).
Polynomials & Terms
- A term is a product of numbers and variables (e.g., ( 2x )).
- A polynomial is a sum or difference of terms (e.g., ( 4x^2 + 3x + 1 )).
- Monomial: 1 term; Binomial: 2 terms; Trinomial: 3 terms; 4+ terms: n-term polynomial.
- The degree of a term is the sum of exponents on its variables.
- The degree of a polynomial is the highest-degree term among its terms.
Collecting Like Terms
- Like terms have identical variables with identical exponents.
- Combine like terms by adding/subtracting coefficients and keeping variable part unchanged.
- Only coefficients are combined; variables and exponents remain the same.
Distributive Property
- Multiply a monomial by a polynomial by multiplying the monomial by each term inside brackets.
- For expressions with multiple brackets, distribute and then combine like terms as needed.
Adding & Subtracting Polynomials
- Treat missing coefficients as 1 or -1 as implied.
- Remove brackets by distributing (if necessary), then collect like terms in descending degree order.
Solving Equations
- To solve, isolate the variable by performing inverse operations, keeping equations balanced (apply the same operation to both sides).
- For two-step equations, first isolate the term with the variable, then solve for the variable.
- With variables on both sides, bring all variable terms to one side and constants to the other before solving.
- Use distributive property to eliminate brackets before solving.
- For equations with fractions, multiply both sides by the common denominator to clear fractions.
- For equations with only fractions, cross-multiplication is a shortcut if both sides are single fractions.
- Word problems: define variables, write expressions, set up an equation, and solve.
Key Terms & Definitions
- Exponent — the number that shows how many times the base is multiplied by itself.
- Base — the number or variable being multiplied in a power.
- Term — a product of numbers and variables.
- Polynomial — an algebraic expression made up of terms joined by addition/subtraction.
- Degree — the highest sum of exponents in a term or polynomial.
- Like terms — terms with identical variables and exponents.
- Distributive property — multiplying a term outside the brackets by each term inside.
Action Items / Next Steps
- Watch Part 2 for Linear Relations and Part 3 for Geometry sections.
- Practice example problems involving each exponent law, polynomial classification, and solving various types of equations.