Algebra 1 - Factoring Polynomials Flashcards

Prime Factorization

• Two methods: Factor tree and Ladder diagram.
• Example: Prime factorization of 40 is (2^3 \times 5) or (5 \times 2 \times 2 \times 2).
• Prime factorization of 17 is simply 17 or (17 \times 1).

Greatest Common Factor (GCF)

• Two methods to find GCF: List the factors or use Prime Factorization.
• GCF of 12 and 16 is 4, derived from their factor lists.
• For monomials, find the GCF by:
  • Writing prime factorizations of coefficients.
  • Aligning common factors.
  • Multiplying the common factors.

GCF Examples

• GCF of (6x^2) and (7y^3) is 1 (no common factors).
• GCF of (18x^2) and (27x^3) is (9x^2).

Properties

• Distributive Property: (a(b+c) = ab + ac).
• Factoring by GCF reverses the distributive property: (xy + xz = x(y+z)).

Factoring Techniques

• Steps for factoring using GCF:
  • Find the GCF of the polynomial terms.
  • Write terms as products using the GCF.
  • Apply the Distributive Property to factor out the GCF.
  • Verify the answer.
• Example of factoring (8x^3 - 4x^2 - 16x) yields (4x(2x^2 - x - 4)).

Factoring by Grouping

• Steps for factoring by grouping:
  • Group terms with common factors.
  • Factor out the GCF from each group.
  • Factor out the common binomial factor.
  • Check solution.
• Example: Factoring (6x^4 - 4x^3 + 12x - 8) results in ((3x - 2)(2x^3 + 4)).

Special Cases

• When factoring two opposing quantities, the sum is 0.
• Example: Factoring (15x^2 - 10x^3 + 8x - 12) results in ((3 - 2x)(5x^2 - 4)).

Trinomials and Polynomial Types

• Standard form for factoring a trinomial: (x^2 + bx + c).
• Example: Factoring (x^2 - 4x - 32) gives ((x - 8)(x + 4)).
• Example: Factoring (20x^2 + 21x - 5) results in ((4x + 5)(5x - 1)).

Definitions

• Polynomial: Combination of terms through addition or subtraction.
• Monomial: One-term expression (number, variable, or product).
• Binomial: Polynomial consisting of two terms.
• Trinomial: Polynomial consisting of three terms.
• Factors: Elements being multiplied in an expression.
• GCF: Largest common factor among terms.
• Zero Product Property: If a product equals zero, at least one factor must be zero.

Test your understanding of factoring polynomials with these flashcards. This quiz includes methods like factor trees and ladder diagrams, as well as finding GCFs and prime factorizations. Perfect for Algebra 1 students looking to reinforce their knowledge.