Unit 1 Chapter 3 Factors and Products By the end of this unit, I should be able to Write a number as a product of its prime factors Explain why 0 and 1 have no prime factors Use prime factorization to: o Determine the greatest common factor (GCF) of two numbers o Determine the least common multiple (LCM) of two numbers o Determine the square and cube root of a number Multiply binomials Multiply polynomials Factor binomials Factor trinomials of the form: o x 2 + bx + c o ax 2 + bx + c o (a + b) 2 o (a b) 2 o a 2 b 2 New vocabulary: prime factorization greatest common factor least common multiple perfect cube cube root radicand radical index factor by decomposition perfect square trinomial difference of squares
Intro to Precalculus and Applied Mathematics Name: Homework Exercises Lesson Questions Done? 1: FACTORS AND MULTIPLES OF WHOLE NUMBERS #3-6, 8-12, 17 2: PERFECT SQUARES, PERFECT CUBES, AND THEIR ROOTS #4-10 3: COMMON FACTORS OF A POLYNOMIAL #7 10, 14, 16 4: MODELLING TRINOMIALS AS BINOMIAL PRODUCTS 5: POLYNOMIALS OF THE FORM xx 2 + bbbb + cc 6: POLYNOMIALS OF THE FORM aaxx 2 + bbbb + cc 7. MULTIPLYING POLYNOMIALS 8. FACTORING SPECIAL POLYNOMAILS # 1 #4, 5, 8, 9,11, 12,14, 15, 21 # 5-6, 9-10, 13, 15, 18 #4-5, 8-10, 13, 15 #4 6, 8, 10-13 Name: Signature: Total
- Factors and Products - APM-20S Name: LESSON 1 FACTORS AND MULTIPLES OF WHOLE NUMBERS Lesson Focus: Determine prime factors, greatest common factors, and least common multiples of whole numbers. Lesson Vocabulary: The prime factorization of a natural number is the number written as a product of its prime factors. Every composite number can be expressed as a product of prime factors. The greatest common factor of two or more numbers is the greatest factor the numbers have in common. The least common multiple of two or more numbers is the least number that is divisible by each number. When a factor of a number has exactly two divisors, 1 and itself, the factor is called a prime factor. For example, the factors of 12 are 1,2,3,4,6,12. The prime factors of 12 are 2 and 3. To determine the prime factorization of 12, write 12 as a product of its prime factors: (2)(2)(3) or (2 2 )(3) To avoid confusing the multiplication symbol with the variable x, we use a dot to represent the multiplication operation, 2 12 2 2 3 or 12 2 3 The first ten prime numbers are 2,3,5,7,11,13,17,19,23,29. Natural numbers greater than 1 that are not prime are composite. Every composite number can be expressed as a product of prime numbers. Example 1 Determining the prime factors of a whole number
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- Factors and Products - APM-20S Example 2 Determining the greatest common factor
- Factors and Products - APM-20S Your Turn To generate multiples of a number, multiply the number by the natural numbers; that is 1,2,3,4,5 and so on. Some multiples of 26 are 26, 52, 78, 104. For 2 or more natural numbers, we can determine their least common multiple. For example, to determine the least common multiple of 4 and 6 list the multiples of both until you find the FIRST one they have in common. 4 = 4, 8, 12, 16, 20, 24. 6 = 6, 12, 18, 24. Notice they have both 12 and 24 in common. We can continue listing them to find more numbers in common. But remember we want the LEAST common multiple. So, the least common multiple of 4 and 6 is 12.
- Factors and Products - APM-20S Example 3 Determining the least common multiple
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- Factors and Products - APM-20S Example 4 Solving problems that involve greatest common factor and least common multiple
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- Factors and Products Practice Problems - APM-20S
- Factors and Products Practice Problems - APM-20S
- Factors and Products - APM-20S LESSON 2 PERFECT SQUARES, PERFECT CUBES, AND THEIR ROOTS Lesson Focus: Identify perfect squares and perfect cubes, then determine square roots and cube roots. Lesson Vocabulary: The square root of a number, n, is denoted by n, is a positive number whose square is n. The cube root of a number, n, denoted by 3 n, is a number whose cube is n. Any whole number that can be represented as the area of a square with a whole number side length is a PERFECT SQAURE. 25 is a perfect square and 5 is its square root. Any whole number that can be represented as the volume of a cube with a whole number edge length is a PERFECT CUBE. The edge length of the cube is the cube root. 216 is a perfect cube and 6 is its cube root. Example 1 Determining the square root of a whole number
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- Factors and Products - APM-20S Example 2 Determining the cube root of a whole number
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Example 3 Using roots to solve a problem - Factors and Products - APM-20S
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- Factors and Products Practice Problems - APM-20S
- Factors and Products Practice Problems - APM-20S
- Factors and Products - APM-20S LESSON 3 COMMON FACTORS OF A POLYNOMIAL Lesson Focus: Model and record factoring a polynomial When we write a polynomial as a set of factors, we factor the polynomial. We say that a polynomial is factored fully when the polynomial cannot be factored further. Factoring and expanding are inverse processes. Example 1 Use greatest common factor with binomials
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Example 2 Factoring Trinomials - Factors and Products - APM-20S
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- Factors and Products - APM-20S Example 3 Factoring polynomials in more than one variable
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- Factors and Products Practice Problems - APM-20S
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- Factors and Products - APM-20S Lesson 4: Modelling Trinomials as Binomial Products Lesson Focus: Factoring polynomials with algebra tiles. This is a large portion of the course and introducing it with a pictorial representation will be critical to your understanding of it! Bare with it!
- Factors and Products - APM-20S
- Factors and Products - APM-20S You Try [Answer: x 2 x 2 or x 2
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- Factors and Products - APM-20S
- Factors and Products - APM-20S You Try [answer: 2x 2 x 1 or 2 x 1 x 1 or 2 x 1
- Factors and Products Practice Problems - APM-20S State the binomial multiplication afterwards
- Factors and Products - APM-20S 2 LESSON 5 - POLYNOMIALS OF THE FORM x bx c Lesson Focus: Use models and algebraic strategies to multiply binomials and to factor trinomials.
Example 1 Multiplying two binomials - Factors and Products - APM-20S
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Example 2 Factoring Trinomials - Factors and Products - APM-20S
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- Factors and Products - APM-20S Example 3 Factoring a trinomial written in ascending order Your Turn
- Factors and Products - APM-20S Example 4 Factoring a trinomial with a common factor and binomial factors
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- Factors and Products Practice Problems - APM-20S
- Factors and Products Practice Problems - APM-20S
- Factors and Products - APM-20S 2 LESSON 6 POLYNOMIALS OF THE FORM ax bx c Lesson Focus: Extend the strategies for multiplying binomials and factoring trinomials. Lesson Vocabulary: Factoring by decomposition is factoring after writing the middle term of a trinomial as a sum of two terms, then determining a common binomial factor from the two pairs of terms formed. Example 1 Multiplying two binomials with positive terms
- Factors and Products - APM-20S Your Turn Example 2 Multiplying two binomials with negative coefficients
- Factors and Products - APM-20S Example 3 Factoring a trinomial by decomposition
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- Factors and Products - APM-20S Example 4 Factoring a trinomial by decomposition
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- Factors and Products Practice Problems - APM-20S
- Factors and Products Practice Problems - APM-20S
- Factors and Products - APM-20S LESSON 7 MULTIPLYING POLYNOMIALS Lesson Focus: Extend the strategies for multiplying binomials to multiplying polynomials. The distributive property can be used to perform any polynomial multiplication. Each term of one polynomial must be multiplied by each term of the other polynomial. Example 1 Using the distributive property to multiply two polynomials
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- Factors and Products - APM-20S Example 2 Multiplying polynomials in more than one variable
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- Factors and Products - APM-20S Example 3 Simplifying sums and differences of polynomial products
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- Factors and Products Practice Problems - APM-20S
- Factors and Products Practice Problems - APM-20S
- Factors and Products - APM-20S LESSON 8 FACTORING SPECIAL POLYNOMIALS Lesson Focus: Investigate some factoring patterns
- Factors and Products - APM-20S Example 1 Factoring a perfect square trinomial
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- Factors and Products - APM-20S Example 2 Factoring Trinomials in two variables
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Example 3 Factoring a difference of squares - Factors and Products - APM-20S
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- Factors and Products Practice Problems - APM-20S
- Factors and Products Practice Problems - APM-20S