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## C . Determining Factors and Products

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**C. Determining Factors and Products**Math 10: Foundations and Pre-Calculus**FP10.1**• Demonstrate understanding of factors of whole numbers by determining the: • prime factors • greatest common factor • least common multiple • principal square root • cube root. • FP10.5 • Demonstrate understanding of the multiplication and factoring of polynomial expressions (concretely, pictorially, and symbolically) including: • multiplying of monomials, binomials, and trinomials • common factors • trinomial factoring • relating multiplication and factoring of polynomials.**Key terms:**• Find the definition of each of the following terms: • Prime Factorization • Greatest Common Factor • Least Common Multiple • Perfect Cube • Cube Root • Radicand • Radical • Index • Factoring by Decomposition • Perfect Square Trinomial • Difference of Squares**1. Factors and Multiples of Whole Numbers**• FP10.1 • Demonstrate understanding of factors of whole numbers by determining the: • prime factors • greatest common factor • least common multiple**1. Factors and Multiples of Whole Numbers**• Two belts are created, one 12 beads long and the second 40 beads long. How many beads long must a belt be for it to created using either pattern?**When a factor of a number has exactly 2 divisors, 1 and**itself, the factor is a Prime Factor • For example, the factors of 12 are 1,2,3,4,6,12. The prime factors are 2 and 3. • To determine the prime factorization of 12, write as a product of its prime factors.**To avoid confusing the multiplication sign with variable x,**we use a dot to represent the multiplication operation. • Example**The 1st 10 prime numbers are:**• 2,3,5,7,11,13,17,19,23,29 • Natural numbers greater than 1 that are not prime are composite**For 2 or more natural numbers, we can determine their**Greatest Common Factor.**To generate multiples of a number, multiply the number by**the natural numbers, that is 1,2,3,4,5, etc. • For example lets find the multiples of 26.**For 2 or more natural numbers, we can determine their Lowest**Common Multiple • When producing multiples for each number the first common one that comes up is the LCM.**Practice**• Ex. 3.1 (p. 140) #3-20**2. Perfect Squares, Cubes and Their Roots**• FP10.1 • Demonstrate understanding of factors of whole numbers by determining the: • principal square root • cube root.**So a perfect cube is a number that can be written as an**integer multiplied by itself three times • Example – 8 is a perfect cube because….**Which number from 1 to 200 represent perfect squares?**• Which represent perfect cubes?**Recall that a perfect square is a number that can be written**as an integer multiplied by itself • Example – 36 is a perfect square because….**Any whole number that can be represented as the area of a**square with a whole number side length is a perfect square • The side length of the square is the square root of the area of the square. • Example**Any whole number that can be represented 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 of the volume of the cube. • Example**Practice**• Ex. 3.2 (p. 14) #1-14, 17 #1-5, 7-18**3. Factors in Polynomials**• FP10.5 • Demonstrate understanding of the multiplication and factoring of polynomial expressions (concretely, pictorially, and symbolically) including: • multiplying of monomials, binomials, and trinomials • common factors**When we write a polynomial as a product of factors, we**factor the polynomial. • The diagrams on the last slide show that there are 3 ways to factor the expression 4m+12**Lets compare multiplying and factoring in arithmetic and**algebra**Practice**• Ex. 3.3 (p. 154) #3-18 #3-5, 9-21**4. Polynomials of the form x2+bx+c**• FP10.5 • Demonstrate understanding of the multiplication and factoring of polynomial expressions (concretely, pictorially, and symbolically) including: • multiplying of monomials, binomials, and trinomials • common factors • trinomial factoring • relating multiplication and factoring of polynomials.**Construct Understanding p. 159**• Pattern – coefficient of c2 is the product of the coefficients in front of c terms • End (lone) coefficient is the product of the two lone coefficients • The coefficient in front of c term is the sum of the two lone coefficients**When 2 binomials contain only positive terms, here are 2**strategies that can be used to determine the product of the binomials. • Algebra Tiles (c+5)(c+3) • Area Model (h+11)(h+5)**These strategies show that there are 4 terms in the product**• These terms are formed by applying distributive property and multiplying each term in the first binomial by each term in the second binomial. • Example (h+11)(h+5)**The acronym, and method used to remember the multiplying**strategy is FOIL. • F – 1st term in each mult. together • O – outside term in each mult. together • I – inside terms in each mult. together • L – last term in each mult. together • Then we add like terms together**When binomials have negative terms it is hard to use algebra**tiles or the area model so we use FOIL to solve.**Factoring and multiplying are inverses of each other.**• We can use this to factor a trinomial.