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Comprehensive Overview of Algebraic Properties and Operations

This guide provides an in-depth exploration of fundamental algebraic properties including the commutative, associative, and distributive properties of addition and multiplication. It covers essential exponent rules, such as the product and power rules, and demonstrates techniques for multiplying monomials and polynomials, including the FOIL method for binomials. Additionally, it delves into polynomial simplification, factorization methods (including prime factor trees and factoring by grouping), and essential concepts like the difference of squares and the sum/difference of cubes.

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Comprehensive Overview of Algebraic Properties and Operations

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  1. 1.2 - Products Commutative Properties Addition: Multiplication:

  2. 1.2 - Products Associative Properties Addition: Multiplication:

  3. 1.2 - Products Distributive Property of Multiplication

  4. 1.2 - Products Product Rule for Exponents If m and n are positive integers and a is a real number, then Examples:

  5. 1.2 - Products Power Rule for Exponents If m and n are positive integers and a is a real number, then Examples:

  6. 1.2 - Products Power of a Product Rule If m, n, and r are positive integers and a and b are real numbers, then Examples:

  7. 1.2 - Products Multiplying Monomials by Monomials Examples:

  8. 1.2 - Products Multiplying Monomials by Polynomials Examples:

  9. 1.2 - Products Multiplying Two Binomials using FOIL First terms Outer terms Inner terms Last terms

  10. 1.2 - Products Multiplying Two Binomials using FOIL First terms Outer terms Inner terms Last terms

  11. 1.2 - Products Squaring Binomials

  12. 1.2 - Products Multiplying Two Polynomials Examples:

  13. 1.3 – Sums and Differences Algebraic Expression - A combination of operations on variables and numbers. Evaluate the following:

  14. 1.3 – Sums and Differences Evaluate the following:

  15. 1.3 – Sums and Differences Simplify each polynomial.

  16. 1.3 – Sums and Differences Simplify each polynomial.

  17. 1.4 - Factorizations A Number as a Product of Prime Numbers Factor Trees

  18. 1.4 - Factorizations A Number as a Product of Prime Numbers Factor Trees

  19. 1.4 - Factorizations A Number as a Product of Prime Numbers

  20. 1.4 - Factorizations + + 10x + 2x 4x2 5 Factoring by Grouping

  21. 1.4 - Factorizations - - 8x + 3x 2x2 12 Factoring by Grouping

  22. 1.4 - Factorizations - + 14x - 10x 35x2 4 Factoring by Grouping

  23. 1.4 - Factorizations Factors of 50 are: 1, 50 5,10 2, 25 Factoring Trinomials

  24. 1.4 - Factorizations 3, 12 4, 9 6, 6 Factors of 36 are: 1, 36 2, 18 Factoring Trinomials

  25. 1.4 - Factorizations Factors of 9 are: 1, 9 3, 3 Factoring Trinomials

  26. 1.4 - Factorizations Product of 2 and 12: 24 Factors of 24 are: 2, 12 1, 24 3, 8 4, 6 Factoring Trinomials Factors of 24 that combine to 11: 3, 8 - - 8x + 3x 2x2 12

  27. 1.4 - Factorizations Product of 12 and 3: 36 Factors of 36 are: 2, 18 4, 9 6, 6 1, 36 3, 12 Factoring Trinomials Factors of 36 that combine to 16: 2, 18 + - 18ab 3b2 12a2 2ab -

  28. 1.4 - Factorizations The Difference of Two Squares Not the difference

  29. 1.4 - Factorizations The Difference of Two Squares

  30. 1.4 - Factorizations The Sum and Difference of Two Cubes

  31. 1.4 - Factorizations The Sum and Difference of Two Cubes

  32. 1.5 – Quotients What is the Rule? Zero Exponent

  33. 1.5 – Quotients Problem: If a is a real number other than 0 and n is an integer, then

  34. 1.5 – Quotients Examples:

  35. 1.5 – Quotients If a is a real number other than 0 and n is an integer, then Examples:

  36. 1.5 – Quotients Examples:

  37. 1.5 – Quotients Practice Problems

  38. 1.5 – Quotients Practice Problems

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