C . Determining Factors and Products

1. C. Determining Factors and Products Math 10: Foundations and Pre-Calculus

2. 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.

3. 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

4. 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

5. 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?

6. Construct Understanding p.134

7. 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.

8. To avoid confusing the multiplication sign with variable x, we use a dot to represent the multiplication operation. • Example

9. 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

10. Example

11. For 2 or more natural numbers, we can determine their Greatest Common Factor.

12. Example

13. 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.

14. 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.

15. Example

16. Example

17. Practice • Ex. 3.1 (p. 140) #3-20

18. 2. Perfect Squares, Cubes and Their Roots • FP10.1 • Demonstrate understanding of factors of whole numbers by determining the: • principal square root • cube root.

19. 2. Perfect Squares, Cubes and Their Roots

20. 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….

21. Which number from 1 to 200 represent perfect squares? • Which represent perfect cubes?

22. 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….

23. 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

24. 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

25. Examples

26. Practice • Ex. 3.2 (p. 14) #1-14, 17 #1-5, 7-18

27. 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

28. 3. Factors in Polynomials

30. 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

34. Lets compare multiplying and factoring in arithmetic and algebra

35. Examples

37. Example

38. Practice • Ex. 3.3 (p. 154) #3-18 #3-5, 9-21

39. 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.

40. 4. Polynomials of the form x2+bx+c

41. 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

42. Strategies – FOIL, Algebra Tiles, Area Model

43. 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)

44. 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)

45. 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

46. When binomials have negative terms it is hard to use algebra tiles or the area model so we use FOIL to solve.

47. Example

48. Factoring and multiplying are inverses of each other. • We can use this to factor a trinomial.