Math 10C Unit 1 Chapter 3 Factors and Products

Chapter Overview

  • Focus on factors, multiples, and polynomials.

  • Important operations and their mathematical properties.

Key Concepts

Determining Factors and Multiples

  • Understanding factors and multiples for numbers up to 100.

  • Identifying prime numbers (only divisible by 1 and itself) and composite numbers (has divisors other than 1 and itself).

  • Knowing definitions for perfect cubes, cube roots, radicands, and radicals.

Additional Polynomial Operations

  • Adding and subtracting polynomials.

  • Multiplying and dividing polynomials by monomials.

Big Ideas

  • Arithmetic operations on polynomials are akin to those on integers.

  • Multiplication and factoring act as inverse processes, representing these visually through area models.

Vocabulary

  • Prime Factorization: Breaking down numbers into prime number products.

  • Greatest Common Factor (GCF): The largest joint divisor among numbers.

  • Least Common Multiple (LCM): The smallest multiple common to a set of numbers.

  • Perfect Cube: A number that is the cube of an integer.

  • Cube Root: A number that multiplies by itself three times to give the original number.

  • Radicand: The number under a radical sign.

  • Radical: The symbol for root operations.

  • Index: Indicates which root is being taken (e.g., square, cube).

  • Factoring by Decomposition: A method of breaking down polynomials into their factors through systematic strategies.

Factors and Products

  • Example: Finding GCF for two numbers (ex: 138 and 198) involves listing their factors and identifying the highest shared one using various methods:

    • Using division,

    • Listing factors,

    • Prime factorization.

Multiples

  • Generates multiples by multiplying the number with natural numbers.

  • LCM is found through shared multiples.

  • Example: Finding the least common multiple for 18, 20, and 30 by listing or through prime methods.

Special Polynomial Factoring Techniques

Perfect Square Trinomials

  • Forms:

    • a2 ± 2ab + b2 => (a ± b)².

  • Recognizing when a trinomial can be factored as a perfect square.

  • Example: 4x2 + 12x + 9 factors to (2x + 3)².

Difference of Squares

  • Recognized as:

    • a² - b² => (a + b)(a - b).

  • Simplifying and identifying differences using square terms.

  • Example: 81 - 36x² = (9 + 6x)(9 - 6x).

Example Problem Solutions

Finding GCF and LCM

  • To find GCF, identify factors for given numbers and select the greatest common one (ex: GCF of 138 and 198 is 6).

  • To calculate LCM, generate a list of multiples until a number appears in both lists simultaneously.

Factorization Techniques

  • Polynomial factorization can involve various approaches (algebra tiles, area models, or algebraically).

  • Identifying common factors from binomials increases efficiency in factorization.

Algebra Tiles and Area Model Techniques

  • Visual tools such as area models can help understand multiplication and factorization tasks, especially with polynomials.

  • Using these representations can deepen understanding of polynomial relationships.

Practice Questions

  • Determine prime factors for several numbers.

  • Using the GCF and LCM in provided scenarios to solve word problems.

  • Multiply given polynomial forms and check solutions with presented methods (algebra tiles, area models).

  • Factor common polynomials, verify through expansion.