Grade 9 Math Review: Algebra, Exponents, and Polynomials

Introduction

• Review of Grade 9 math concepts in under 60 minutes
• Ideal for exam review or course preview
• Visit jensenmath.ca for in-depth explanations and proofs
• Video divided into three main units: Algebra, Linear Relations, Geometry

Algebra

Exponent Laws

• Power: Base (e.g., 5) raised to an exponent (e.g., 2) means multiplying the base by itself as many times as the exponent value (e.g., 5² = 5 * 5)
• Product of Powers Rule: Keep the base, add the exponents (x^a * x^b = x^(a+b))
• Quotient of Powers Rule: Keep the base, subtract the exponents (x^a / x^b = x^(a-b))
• Power of a Power Rule: Keep the base, multiply the exponents (x^(a^b) = x^(a*b))
• Power of a Quotient: Apply the exponent to both the numerator and the denominator (a/b)^x = a^x / b^x
• Power of a Product: Apply the exponent to each factor (ab)^x = a^x * b^x
• Zero Exponent Rule: Any base with an exponent of 0 is 1 (x^0 = 1)
• Negative Exponent Rule: Rewrite with positive exponent using the reciprocal (x^-a = 1/x^a)

Examples and Practice

• Simplifying expressions using the above rules
  • Examples: 4^3 * 4^5 / 4^2 = 4^6
  • (-2/3)^3 = (-2)^3 / 3^3
  • 5^4 * 5^-7 = 1 / 5^3
  • a^3 * a^4 * a^5 = a^12
  • 4(x^2y^5)^3 = 64x^6y^15
  • 36z^3 / 27z^6 = 4 / 3z^3
  • (8b^3d)(4d^2) = 32b^3d^3
  • (x+4)/2 - (x+1)/4 = 3/8

Polynomials

• Term: Product of numbers and variables (e.g., 2x)
• Polynomial: Expression with one or more terms separated by addition or subtraction
  • Monomial: 1 term
  • Binomial: 2 terms
  • Trinomial: 3 terms
  • Four-term polynomial, etc.
• Degree of a Term: Sum of the exponents on all variables
  • Example: 3x^2y has degree 3 (2+1)
• Degree of a Polynomial: Highest degree term
  • Example: 4x^5 + 3x^4 + 2 has degree 5

Collecting Like Terms

• Like Terms: Same variables with the same exponents
  • Example: 3x^2 and 5x^2 are like terms
• Combine like terms by adding/subtracting coefficients, keeping the variables/exponents the same
  • Example: 2x + 4x = 6x
• Practice
  • Combine and simplify: 3x + 5x = 8x
  • 7y - 9y = -2y
  • 3a^2b + 4a^2b = 7a^2b

Distributive Property

• Apply the term outside the parentheses to each term inside
  • Example: a(b + c) = ab + ac
• Simplifying expressions
  • 5(4x + 2) = 20x + 10
  • 3(x - 1) + 2(x + 4) = 3x - 3 + 2x + 8 = 5x + 5

Solving Equations

• Balanced Method: Keep equation balanced by doing the same operation on both sides
  • Example: x + 4 = 7 -> x = 3
• Isolating the Variable: Move terms to isolate the variable
  • Subtract/add terms: g - 5 = -3 -> g = 2
  • Multiply/divide terms: 5u = -20 -> u = -4
• Two-Step Equations
  • Isolate term with the variable first, then solve
  • Example: 7y + 8 = 15 -> y = 1
• Equations with Fractions
  • Multiply both sides by the denominator to eliminate the fraction
  • Example: c/3 = 2 -> c = 6
  • Cross Multiplication: (a/b) = (c/d) -> ad = bc

Word Problems

• Translating word problems into algebraic equations
  • Example: Natasha has 250 more tabs than Kristen. Together they have 880 tabs.
  • Let x represent Kristen's tabs, then Natasha has x + 250
  • Equation: x + (x + 250) = 880 -> x = 315, Natasha = 565

Conclusion

• Review important algebra concepts
• Watch parts 2 and 3 for Linear Relations and Geometry