Grade 9: Algebra Flashcards

Algebra Basics

• Algebra involves using letters and symbols to represent numbers and express their relationships.
• An expression combines numbers, symbols, and variables, but does not have an equal sign.

Key Terms

• Coefficients: The numerical part placed before a variable in an algebraic term.
• Variables: The unknown elements in an expression that can have exponents.
• Constants: Numbers that do not change and are not accompanied by variables.
• Terms: The distinct components that make up an expression, either variables, constants, or products.

Terms and Degrees

• Degree of a Term: Calculated by summing the exponents of all variables within that term.
• Polynomial: An expression containing multiple terms connected by addition or subtraction.
• Types of polynomials include:
  • Monomial: Contains one term.
  • Binomial: Contains two terms.
  • Trinomial: Contains three terms.
• Degree of Polynomial: The term with the highest degree within the polynomial dictates the overall degree.

Factoring and Common Factors

• Factor: A number that divides another number without leaving a remainder.
• Factoring: The process of identifying multiple components that can be multiplied to yield a particular expression.
• Common Factor: The largest factor shared among terms in an expression, often determined by the Greatest Common Factor (GCF).

Operations with Terms and Polynomials

• Identifying degrees:
  • Utilize alphabetical ordering to identify the degree of terms, for instance, in 7xy, the degree is 2.
• Collecting Like Terms: Group terms that share the same variables and exponents to simplify expressions.
• Multiplying Monomials: Multiply coefficients and add exponents for like variables.
• Dividing Monomials: Divide coefficients and subtract exponents for like variables.

Power Rules and Polynomials

• Applying the power of a power rule involves raising each component to the appropriate exponent, e.g., ((4c^3d^4)^2 = 16c^6d^8).

Adding and Subtracting Polynomials

• Combine like terms by aligning similar variables and coefficients, e.g., simplifying ((3a - 2b + c - 10) + (6a + c - 4) - (14b - 3c)) results in (9a - 16b + 5c - 14).

Multiplying and Dividing Polynomials

• Multiplying Polynomials: Use the distributive property (rainbow method) to expand polynomials such as (3x(4x^2y + 2xy - 7) + 2x^3(3x - 5xy + 2)).
• Dividing Polynomials: Similar to monomial division but considers polynomial structure.

Expanding and Common Factoring

• Expanding Binomials: The double rainbow method expands products such as (-5(2x - 1)(5x^2 - 3x - 2)) into a polynomial expression.
• Common Factoring: Reverse expand by factoring out the GCF from an expression to simplify further, e.g., (9x^2y^3 + 12xy^4 - 18x^3y^2 + 3xy^2) becomes (3xy^2(3xy + 4y^2 - 6x^2 + 1)).