Unit 3 Factors & Products

Transcription

1 1 Unit 3 Factors & Products General Outcome: Develop algebraic reasoning and number sense. Specific Outcomes: 3.1 Demonstrate an understanding of factors of whole number by determining the: o prime factors o greatest common factor o least common multiple o square root o cube root 3. Demonstrate an understanding of the multiplication of polynomial expressions (limited to monomials, binomials, and trinomials), concretely, pictorially, and symbolically. 3.3 Demonstrate an understanding of common factors and trinomial factoring, concretely, pictorially, and symbolically. Topics: Prime Factorization, Greatest (Outcome 3.1) Common Factor, Lowest Common Multiple Multiplying Polynomials (Outcome 3.) Common Factors (Outcome 3.3) Factoring by Grouping (Outcome 3.3) Factoring by Decomposition (Outcome 3.3) Factoring Strategies (Outcome 3.3)

2 Prime Numbers: Unit 3 Factors & Products Prime numbers are numbers that have exactly two divisors, 1 and itself. In other words, these are numbers that can only be divided evenly by 1 and itself. Ex) 7, 11, 19 **The number 1 is not considered prime as it only has one number it can be divided by evenly. Ex) List all of the prime numbers up to 0. Composite Numbers: Composite numbers are numbers that have more than two divisors or are numbers that can be evenly divided by more than two numbers. Ex) 1, 6, 45

3 3 Prime Factorization: The prime factorization of a natural number is the number written as a product of its prime factors. Ex) Determine the prime factorization of the following numbers

4 4 Greatest Common Factor: The greatest common factor of two or more numbers is the largest number that each can be divided by. Ex) Determine the greatest common factor for each set of numbers. a) 1 and 0 b) 138 and 198 c) 16, 144, and 630

5 5 Least Common Multiple: The least common multiple is the smallest number that can be divided evenly by two or more given numbers. Ex) Determine the least common multiple of each set of numbers. a) 14 and 1 b) 18, 0, and 30 c) 8, 4, and 63

6 6 Ex) What is the side length of the smallest square that could be tiled with rectangles that measure 16 cm by 40 cm? Assume the tiles cannot be cut. Ex) What is the side length of the largest square that could be used to tile a rectangle that measures 16 m by 40 m? Assume the squares cannot be cut.

7 7 Prime Factorization / Greatest Common Factor / Lowest Common Multiple Assignment: 1) List the first 6 multiples of each number. a) 6 b) 13 c) d) 31 e) 45 e) 7 ) Express each number as a product of its prime factors. a) 40 b) 75 c) 81 d) 10 e) 140 f) 19 g) 45 h) 160 i) 96

8 8 3) Explain why the numbers 0 and 1 have no prime factors. 4) Determine the greatest common factor for each set of numbers. a) 45, 84 b) 64, 10 c) 81, 16 d) 180, 4 e) 160, 67 f) 0, 860 g) 150, 75, 40 h) 10, 960, 1400 i) 16, 10, 546, 714 j) 0, 308, 484, 988

9 9 5) Determine the lowest common multiple for each set of numbers. a) 1, 14 b) 1, 45 c) 45, 60 d) 38, 4 e) 3, 45 f) 8, 5 g) 0, 36, 38 h) 15, 3, 44 i) 1, 18, 5, 30 j) 15, 0, 4, 7 6) When is the product of two numbers equal to their least common multiple?

10 10 7) Use greatest common factors or prime factorization to simplify the following fractions. a) b) c) d) e) f) ) A developer wants to subdivide this rectangular plot of land into congruent square pieces. What is the side length of the largest possible square? 300 m 400 m

11 11 Multiplying Polynomials: Monomial Polynomial When multiplying a monomial to a polynomial, the monomial is distributed into the brackets Ex) 3x(5x + 9) ab (14a 5a b 3 + b) Ex) Expand the following. a) 3x( 5x 7x) b) 5x 3 ( 9x 4 3x 3 + x 1) c) 7xy ( 3x 3 y + x 4 6y + x) d) 10y 4 ( 6x y 3 + xy)

12 1 Polynomial Polynomial When multiplying one polynomial to another, each term of one polynomial must be multiplied to each term of the other polynomial. Once fully expanded look for like terms in order to simplify the answer. Ex) (x + 5)(x 7) Ex) (x 4)(x + 1) Ex) ( x + 5x + 3)( x 8)

13 13 Ex) Expand and simplify the following. a) (a + 8)(a 3) b) (3x y)(x + 3y) c) (x + 4)(3x + 7x 10) d) (x + 8x )(3x 5x + 6)

14 14 e) (x + 3)(x 1)(4x + 5) f) ( 3xy + 5)( 7x xy + y ) g) ( x + 3) ( x 3 + 5x 3) h) 7a ( a + 1)( a 3a + 5)

15 15 i) ( 4x + 6)( x ) + ( x + 3)( x 5) j) ( 4x 1) ( 7x + 3)( x 5) k) ( 3x 4y)( x + y) ( 5x + )

16 16 Multiplying Polynomials Assignment: 1) Expand and simplify the following. a) ( t + t + )( t + 3t + 1) b) ( w + 3)( w + 4w + 7) c) ( x y)( 3x y) + + d) ( 4x + 3)( 3x 4x + 1) e) ( 5x+ 4)( 5x 4) f) ( 3t + 6) g) ( a + 5b + 3)( a + 4b) h) ( b + 9b )( b 1)

17 17 i) ( d + d + 1)( d 6d + 3) j) ( x 4)( x + 4x + 16) k) ( a b)( a 4b c) + l) ( a 5)( a + 3a 4) ) Expand and simplify the following. a) a( a 1)( 3a + ) b) 5x ( x 1)( 4x 3) c) b( b c)( b + c) d) ( 3x + 4)( x 5)( x + 8)

18 18 e) ( 5a 3)( a 7) f) ( x+ 5y) 3 g) ( x + y z) 3 3) Expand and simplify the following. a) ( 3x + 5)( x + ) + ( 3x + 7)( x + 6) b) ( x + 3)( 5x + 4) + ( x 4)( 3x 7)

19 19 c) ( 4y 5)( 3y + ) ( 3y )( y + 1) d) ( 3m + 4)( m 4n) + ( 5m )( 3m 6n) e) ( 3x ) ( x 6)( 3x 1) f) ( a + 1)( 4a 3) ( a )

20 0 4) Determine a simplified polynomial expression that represents the area of the shaded region for each figure given below. a) 5x + 8 x + 5 3x 6x + b) x x x + 1 x

21 1 Common Factors: When factoring a polynomial, identify the greatest common factor (largest term that each term in the polynomial can be evenly divided by) and divide this out from each term. The result is written as the product of the greatest common factor and a polynomial. Ex) Factor the following by removing a common factor. 15x 5 1x + 18x 4x 3 5 0a b 15a b + 10a b Ex) Factor the following. a) 6a 4 + a b) 8xy 14x y 3 4

22 c) t t d) 8a + 1a 0a 5 4 e) 3 1x y 0xy + 16x y f) 10x yz + 1x y z g) 100a bc + 15ac 300a bc 150b c h) 3x y z + 16xy z 8xy z i) 5x y 6x yz + 4y z + 8x y z 3 3

23 3 Common Factors Assignment: 1) Factor the following by removing a common factor. a) 5y + 10 b) 3 4x + 14x c) 9a b 1a b 3 3 d) 3x 1x 6 + e) 10a 6 1a f) 4 3 6x + 7x 8x g) 3a 13a 1a 4 3 h) 3 3 4x y + 30x 1y i) 14ab c 35a bc + 1a b c 3 4 j) 4 3 9a b + 3a b 8a b k) 7x y + 14x y 1xy 3 3 3

24 4 ) Simplify each expression by combining like terms, then factor. a) x x x x b) 1a 4a 3 + 4a 13 c) y 5y + y y y 1y d) 7xy 11x y + 18x y + 5x y + 5xy

25 5 Factoring by Grouping: Factoring by grouping can be used to factor polynomials with 4 terms. Ex) x + 8x + 3x + 1 = x + 8 x + 3x + 1 Group the terms = x x x + 4 Remove a common factor ( ) ( ) from each group. The bracketed sections should be the same. = x+ 4 x+ 3 Factor out the common ( )( ) bracketed binomial from each group. Ex) Factor the following by grouping. a) m 10m 3m 15 + b) 10 5 a a + b b c) x 3xy xy + 6y d) ac + 4ad + bc + 8bd

26 6 e) x x x f) 3 x x x g) 9ac + 3a + 3bc + b h) 5x 30x 15x + 90 i) x x xy y x j) ac ab a bc b b

27 7 Factoring by Grouping Assignment: 1) Factor the following using the method of grouping. a) 10x x 15x b) ab + a 4b 8 c) 6x 3x 4x + d) 9x + 15x + 15x + 5 e) 8a + 4ab + 6ab + 3b f) 7x + x y + 4y + 8 g) 5x 10x 10x 4 + h) 4 4a 8a + 5a 10

28 8 ) Factor the following by first removing a common factor and then using the method of grouping. a) 3x 1x 6x 4 + b) 50a + 10ab b c) x + x + x + x d) 9a b 7a b + ab 3ab 3 3 e) x 6x 6x + x f) 15ac + 0ad + 30bc + 40bd g) x y + x y xy xy h) 30a b + 6ab 75ab + 15b

29 9 3) Factor the following using the method of grouping. a) 3 6x x + 15x 5x 1x + 4 b) a ab a ab b b c) 1x 10x + 6xy 5y 4x + 35

30 30 Factoring by Decomposition: Factoring using the method of decomposition is typically used to factor trinomials. Essentially it is a method that turns a trinomial (3 terms) into a polynomial of 4 terms that can then be factored by grouping. Ex) 3x + 17x = 17 = 30 Find two numbers that add up to the middle coefficient (17) and multiply to the first times the last coefficient (30). 3x + 15x + x + 10 Use these numbers to break up the middle term. 3x x x + 5 Complete using the method ( ) ( ) ( x 5)( 3x ) + + of factoring by grouping. Ex) Factor the following using the method of decomposition. a) 6a 7a 0 b) a + a 1

31 31 c) 8x + 14xy + 3y d) x + 11x e) 8a 34a 9 + f) x 5x 1 g) x + x 48 h) 15x 16x+ 4

32 3 i) 1x 13x 4 + j) a 4 + 5x 36 k) 4x 9 l) 10a + 1a 10 m) 90x 45x 10 + n) x 81

33 33 Factoring by Decomposition Assignment: Factor the following using the method of decomposition. 1) 14x 3x 3 + ) 10a + 3ab b 3) 3a 13a ) x + 13x 30 5) 49x 1 6) a 9ab 5b 7) 11a 8a 3 + 8) x 10x 75

34 34 9) 9x 18x ) t ) 5x 17x 1 1) x + 14x ) 1a 5ab b 14) 6x + 3x ) 100a 9 16) x + x+ 60

35 35 Factoring Strategies: Factor the following. 8a 35a xy xy y 1a b + 1a b 7ab x + 39x 30 Strategy:

36 36 Ex) Factor the following. x + 10x+ 1 a + 6a 16 x 3x 40 x + 5x 36 t 16t+ 48 x 6x 40 Strategy:

37 37 Ex) Factor the following. x 16 4a 5 9t 49 x 100 x x 11 Strategy:

38 38 Ex) Factor the following. a) x 4x 1 b) x + 3x 14 c) 8x 37x 15 + d) 36x 49 e) ab + a + b + b f) b + 11b+ 4 g) 4a 81b h) x + 1x 3

39 39 Factoring Strategies Assignment: Fully factor the following polynomials. 1) x + 4x 3 ) 1x 3x+ 10 3) 3 a 6a 7a 4) 6ab + 4a + 6b + 4 5) 16a b 6) 8x 6x 7 7) 3 x x x ) t 15t+ 54

40 40 9) 4 x 16 10) 3x y + 15x y + 75xy 3 11) 1a + 7a 10 1) b + b ) a 64 14) x y 10xy 4y 15) x 4 10x ) xy 5

41 ) 15a b 0ab + 40a b 18) 4t 14t 3 19) x ) 4b + 8b 7 1) 4x 9y ) 4x + 1x+ 9 3) x 13x ) ( x + 3) 5 4

42 4 Answers Prime Factorization / Greatest Common Factor / Lowest Common Multiple Assignment: 1. a) 6, 1. 18, 4, 30, 33 b) 13, 6, 39, 5, 65, 78 c), 44, 66, 88, 110, 13 d) 31, 6, 93, 14, 155, 186 e) 45, 90, 135, 180, 5, 70 f) 7, 54, 81, 108, 135, 16. a) 5 b) c) d) 3 5 e) 5 7 f) g) h) 5 i) 3 4. a) 3 b) 8 c) 8 d) 4 e) 3 f) 0 g) 5 h) 40 i) 4 j) 4 5. a) 84 b) 315 c) 195 d) 798 e) 1440 f) 364 g) 340 h) 580 i) 900 j) a) m b) c) d) Multiplying Polynomials Assignment: 4 61 e) f) a) d) g) i) l). a) d) f) g) 3. a) d) 4. a) 4 3 t t t t b) 3 w 11w 6w c) 6x + 5xy + y 3 1x 7x 8x + 3 e) 5x 16 f) 9t + 36t a + 13ab + 0b + 3a + 1b h) b + 117b 13b d 10d 5d + 3 j) x 64 k) a + ab ac 4b + bc 4 3 a + 3a 9a 15a a + a 4a b) 40x 50x + 15x c) 4b + b c bc 3 3 6x + x 18x 160 e) 0a 15a + 39a x + 60x y + 150xy + 15y x + 3x y 3x z + 3xy + 3xz 6xyz + y 3y z + 3yz z x 41x b) 13x 4x c) 18m 4mn m 4n e) 3x + 8x f) 7x + 43x+ 16 b) x + x 6y 6y 8 7a + a 7

43 43 Common Factors Assignment: 1. a) 5( y + ) b) x ( + 7x) c) 3a b ( 3b 4a) d) 3( x 4x ) e) ( 5a 3 6a ) f) x( 6x x ) g) a ( a + 1a) h) 6( 4x y 3 + 5x y 3 ) i) 7abc( b 5ac + 3a b 3 c) j) a b( 9a + 3a 8) k) 7xy ( x y + x 3y) x b) 8( a 3a ) c) y( 4y + 3y + 5). a) 4( 1) d) 6xy( y x + 3xy) Factoring by Grouping Assignment: 1. a) ( 5x+ 1)( x+ 3) b) ( b+ )( a 4) c) ( x 1)( 3x ) d) ( 3x + 5) e) ( a b)( 4a 3b) y + 7 x + 4 g) ( 5x ) h) ( a )( 4a + 5) + + f) ( )( ). a) 3( x+ 7)( x ) b) 10( 1+ b )( 5a 3) c) x( x + 5)( x + 6) d) ab ( b 3)( 9a + b) e) x ( 3x 1) f) 5( 3c + 4d )( a + b) g) 3xy( y + 3)( 3x ) h) 3b ( 5a 1)( a + 5) 3x 1 x + 5x 4 b) ( 5a + 4b )( a b) c) ( 6x 5)( x + y 7) 3. a) ( )( ) Factoring by Decomposition Assignment: 1. ( 7x 1)( x 3). ( 5a b)( a + b) 3. ( 3a+ )( a 5) 4. ( x+ 15)( x ) 5. ( 7x 1)( 7x+ 1) 6. ( a + b)( a 5b) 7. ( 11a 3)( a+ 1) 8. ( x 15)( x+ 5) 9. ( 3x 4)( 3x ) 10. ( t 5)( t + 5) 11. ( 5x+ 3)( x 4) ( x + 6)( x + 8) 13. ( 3a b)( 4a + b) 14. ( 6x+ 5)( x+ 3) 15. ( 10a 3)( 10a+ 3) 16. ( x+ 5)( x+ 6)

44 44 Factoring Strategies Assignment: 1. ( x+ 8)( x 4). ( 3x )( 4x 5) 3. a( a 1)( a + 6) 4. ( 3b+ )( a+ 1) 5. ( 4a b)( 4a + b) 6. ( 4x+ 1)( x 7) 7. ( x+ )( x 6) 8. ( t 9)( t 6) 9. ( x + 4)( x + )( x ) 10. 3xy ( x + 5x + 5) 11. ( 4a+ 5)( 3a ) 1. ( b+ 5)( b+ 6) 13. ( a 8)( a+ 8) 14. y( x 1)( x + ) 15. ( x 8)( x ) xy 5 xy 5 5ab 3a 4b + 8a 18. ( 6t+ 1)( 4t 3) 16. ( )( + ) 17. ( ) 19. ( x 7) 0. 4( b+ 9)( b ) 1. ( x 3y)( x + 3y). ( x + 3) 3. ( x 3)( x + 3)( x )( x + ) 4. ( x )( x+ 8)