# Unit: Polynomials and Factoring

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1 Name Unit: Polynomials: Multiplying and Factoring Specific Outcome 10I.A.1 Demonstrate an understanding of factors of whole numbers by determining: Prime factors Greatest common factor Least common multiple 10I.A.3 10I.A.4 10I.A.5 Demonstrate an understanding of powers with integral and rational exponents Demonstrate an understanding of the multiplication of polynomial expressions Demonstrate an understanding of common factors and trinomial factoring Comments : Page 1

2 Naming Polynomials Polynomials by number of terms Monomials (1 term) Binomials ( terms) Trinomials (3 terms) Polynomial with 4 terms Polynomial with 5 terms Etc. Review Lesson on Polynomials Polynomials by highest degree Constant (no variable visible) Linear (variable with exponent of 1) Quadratic (variable with exponent of ) Cubic (variable with exponent of 3) Quartic (variable with exponent of 4) Quintic (variable with exponent of 5) Sixth degree polynomial (variable with exponent of 6) Etc. Collecting and Combining Like Terms When collecting like terms, they must have the same variable and same exponent. Add or subtract the coefficients (NOTE: the + and signs belong to the term behind them). Examples: Simplify the following a) 6xy + 4x + 8xy b) x 3y 6x + y c) 3x + 5x 7y 8x + 8y Page

3 Adding and Subtracting Polynomials Adding 1) Remove the brackets ) Simplify by collecting like terms 3) Combine like terms 4) Arrange in alphabetical order and then from highest to lowest power Example 1: Simplify (x + 4x + 3) + (x + 5x + 1) Subtracting 1) Remove the first set of brackets (unless there is a number in front of it other than +1). NOTE: a subtraction sign is similar to -1. ) Multiply the negative one through the second bracket (take the opposite for each term in the second set of brackets) and remove the brackets. 3) Collect like terms 4) Combine like terms Example : Simplify (4x 5x + 7) (3x + x 4) Page 3

4 Multiplying and Dividing Monomials Multiplying (coefficient coefficient)(variable variable) remember to add exponents when multiplying powers Examples: Find the product a) (30x)(4y) b) -5x (-3yz) c) 4(3x y + 7) use the distributive property to expand each expression multiply each term inside the brackets by the term outside the brackets Dividing divide each term of the polynomial by the monomial (note example 3) remember to subtract exponents for common variables Examples: Simplify a) 1x 3 b) -18a 3 b 5 c 7 6ab c 4 c) 8x 5-16x 4 + 4x x 3 x Assignment: Adding and Subtracting Polynomials handout Multiplying and Dividing Monomials handout Page 4

5 Lesson 1: Multiplying Polynomials (Part 1) Multiplying Monomial by Polynomial: Distribution Method Examples: Multiply the following polynomials a) 5y (x - y) b) 4y(y + 3y - 1) Multiplying Binomial by Binomial: FOIL Method A technique for multiplying two binomials is using the F.O.I.L. method. The letters F. O. I. L. stand for,,, We always multiply these terms. Steps : 1) Identify the first term in each bracket and them together. ) Identify the most outside terms of the expression and multiply them together. 3) Identify the most inside terms of the expression and multiply them together. 4) Identify the last term in each bracket and multiply them together. 5) Simplify by collecting like terms. Examples: Multiply the following polynomials c) (x +)(x + 5) First Outer Inner Last Page 5

6 Multiply the following polynomials d) (x + 6)(x + 8) e) (x - y)(3x + y) f) (x - y)(x + y) Binomial Squared: g) (x + 5) *This is the same as (x + 5)(x + 5) h) (x - y) Page 6

7 Assignment #1 FOIL Multiply the following binomials using FOIL: 1) x 3 x 4 ) x x 5 3) x 1 3x 4) 4x 3 x 5) x 4 6) x 3 3x 4 7) x 4 x 4 8) t 5 t 4 9) 3w w 9 10) z z Page 7

8 11) a b 1) 5e 5 6e 1 13) x x 1 14) x 5 15) x 7y x 3y 16) x 6y x y 17) x 4y 18) x 8 x 7 19) x 3y x 4y 0) x 1 x 10 1) x 8y ) x 8 Page 8

9 Lesson : Multiplying Polynomials (Part ) Multiplying Binomial by Trinomial: Distribution Method Examples: Multiply the following polynomials a) (y - 3)(y - 4y + 7) b) (x - 1)(x + 5x - 3) Examples: Expand or simplify the following polynomials a) 3(x - 1)(x - 3) b) (5a + 4) + (a 1)(a + ) (a 3) Page 9

10 Assignment # Distribution Method Multiply the following polynomials 1) d 3 d 5d ) 4s 5 s 9s 1 3) 3c 6 c 4c 7 4) k 5k k 7 5) 5y y y y 6 6) r 5r 3 3r 4r 5 7) c c 4 c c 7 Page 10

11 Simplify the following polynomials: 1) 4 5y 3 y 3 3y 1 ) 3a 9 a 5 4a 7 6a 3 3) d e 3d 5e 6d 5e d 4e 4) 5n 4 n 7 8n 6 5) 3w w 4 w 5w 6 6) 4t 5s t 3s 5t s 7) 3a 7 4a 3 a 8) b 3b 6 b 3 9) x y x 4y x y 3x y 10) 4 6a c a 3c a c Page 11

12 Lesson 3: Greatest Common Factor and Prime Factors The greatest common factor (GCF) is the largest factor shared by two or more terms. This is the largest number that divides evenly into two or more numbers. In order to determine the GCF of or more numbers, you must begin by listing all the factors of those numbers. When a factor of a number has exactly two divisors, one and itself, the factor is a prime factor. For example, The factors of 1 are 1,, 3, 4, 6, and 1. The prime factors of 1 are 1,, and 3. To determine the prime factorization of 1, write 1 as a product of its prime factors: x x 3, or ² x 3 NOTE: o The first 10 prime numbers are:, 3, 5, 7, 11, 13, 17, 19, 3, and 9 o Natural numbers greater than one that are not prime, are composite. Example 1: Write the prime factorization of Begin by using a Factor Tree. Page 1

13 Example : List the factors of each of the following numbers. Then, identify the GCF. a) 15 and 30 b) -4 and -48 Example 3: Determine the greatest common factor of 4xy and x²y. Example 4: Determine the greatest common factor of the following sets of terms: 18x²yz, 7x²y²z, 9x²y² Page 13

14 Least/Lowest Common Multiple The least/lowest common multiple (LCM) is the smallest multiple shared by two or more terms. To generate multiples of a number, multiply the number by the natural numbers; that is, 1,, 3, 4, 5, and so on. For example, some multiples of 6 are: 6 1 = 6 6 = = 78 For two or more natural numbers, we can determine their least common multiple. Example 5: Determine the least common multiple of 18, 0, and 30. Example 6: Mei is stacking toy blocks that are 1 cm tall next to blocks that are 18 cm tall. What is the shortest height at which the two stacks will be the same height? Page 14

15 Assignment#3 Prime Factors, LCM s and GCF s 1. Two ropes are 48 m and 3 m long. Each rope is to be cut into equal pieces and all pieces must have the same length that is a whole number of meters. What is the greatest possible length of each piece? a) List the factors of both numbers. b) State the Greatest Common Factor (GCF).. Complete each factor tree and write the prime factorization for each number. a) 144 b) 600 c) 5 d) 900 Page 15

16 3. Find the GCF for the following: a. 44 and 70 b. 36 and Find the Lowest Common Multiple (LCM) of the following: a. 1 and 30 b. 16 and Hamburger patties come in packages of 8. Buns come in packages of 6. What is the least number of hamburgers that can be made with no patties or buns leftover? Page 16

17 6. List the prime factors of the coefficients and the variable for each term and find the Greatest Common Factor between the two terms. a. 6x, 1x d. 18xy z, 4x y 3 z b. 0c d 3, 30cd e. 5m 3 n, 0mn c. 4b c 3, 6bc 7. Determine the GCF of the following sets of terms: a. 14a, 1b b. -5n, -10n c. 3rs, 7t d. 1f g 3, 16fg, 3f 3 g e. -15d e 3, -30cd e, -45cde f. -18j 3 k, 7j kl, 36j k l Page 17

18 Lesson 4: Common Factoring Factoring is the process of. The better you are at multiplying, the better you will be at factoring. Multiplication Factoring 5x(x y) = 5x 10xy 5x 10xy = 5x(x y) (x 3)(x + 5) = x + x 15 x + x 15 = (x 3)(x + 5) 1) Common Factoring: When factoring, begin by looking for the GCF. It could contain a number, a variable or both. Place this greatest common factor in front of parentheses, with the remaining polynomial the parentheses. Once this is done, the same number of terms as in the original question should be inside. (i.e. a leaves a.) Examples: Factor the following: i) 4x + 8 = ( ) ii) 8xy 3y = GCF Remaining factor iii) 7n 49n = iv) 15w 3 + 5w = iv) b b r 3 c = vi) 1n 3 16n + 3n = vii) 3x 3 6x y + 9xy = Factoring can always be quickly and easily checked by the polynomials together to see if the product is the original polynomial. Page 18

19 Assignment#4 Factoring Binomials Factor the following polynomials. a. 6s +30 b. 4t +8 c. 5a 5 d. 16r 1 r e. 7xy 14xy 49xz f. 3c 9c 7d 3 f. 15w 5 w h. 4a 6a 3 i. 10x y 50xy j. g 4 g k x y x y xy l. r 6r s 4rs 3 Page 19

20 Lesson 5: Factoring ax + bx + c (leading coefficient) Use this method anytime there is a in front of your x which cannot be factored out. Product-Sum-Factor (PSF) Method Factor: 3x + 17x + 10 ALWAYS begin factoring by checking for common. From the remaining trinomial, calculate the product of the and coefficients. List all of the (List them as pairs) of this product. From these pairs of factors, identify which pair, when added together, result in the middle term of the trinomial. Re-write the trinomial with the original first and last terms. In between these two terms, insert two new terms using the pair of coefficients from step. Now the first two and the last two terms. (Notice how a common factor emerges.) Now brackets. again. Collect the common You can your answer by expanding (multiplying using FOIL) Page 0

21 Examples: Factor the following fully, if possible: 1. 3y 10y + 8 =. 8a + 18a 5 = 3. 15x + 48x + 36= 4. c + c 3 = 5. 5x 0xy + 0y = Page 1

22 6. 6b 4 + 7b 10 = 7. d 3 + 7d 30d = 8. 10g 3gh h = 9. 6k + 14km 1m = Page

23 Assignment #5 Factoring with leading coefficients. Factor the following trinomials: 1. x 3x 1. 3x 5x 3. 6x 13x 6 4. x 5x x 18x x 17x 8 7. r 11r l 11l w 9w b 8b Page 3

24 11. y 5yz 6z 1. 1a 19a f 7 f r r b 6b m 17mn 3n 17. 9g 9gf f 18. 6l 3l 4 Page 4

25 Lesson 6: Trinomial Factoring (leading coefficient of 1) Trinomials will factor to brackets. Example: x + 5x + 6 Steps: ALWAYS factor out any Identify the terms/variables first. of the last term of the trinomial. Next, determine which of factors either up to or to get the middle term Therefore, x + 5x + 6 factors to ( )( ) Examples: Factor the following trinomials fully, if possible: 1. x + 9x y y b 6b 4. a 4a - 60 Page 5

26 5. x + 8xy + 16y 6. p 4 p x + 8x x - 7x x 3-18x + 7x 10. x yz 3-10xyz 3-48z 3 y Page 6

27 Assignment #6 Factoring Trinomials Factor the following trinomials: 1. y 8y 1. x 10x 1 3. a 19a m mn 4n 5. b 19b g 10g 4 7. n 15n 6 8. c 15c s 7st 10t 10. f 6 f 1 Page 7

28 Lesson 7: Factoring Difference of Squares A) Perfect Square Binomials: ax - by A difference of squares has 3 main features: 1. The first term is a perfect.. The second term is a square. 3. They are separated by a sign. Example: x 16y The term is absent because it is. Eg. x - 0xy - 16y Factoring a perfect square binomial results in two similar binomials, that differ only in the sign. To factor a difference of squares: Remember to ALWAYS begin factoring by looking for a factor. The first term of the binomials comes from the square of the term. The term of the binomials comes from the square of the second term. Place a sign in one parentheses and a in the other. Check the result by using F.O.I.L. Example from above: x - 16y = ( )( ) Examples: 1. x x 3-48x. 5b - a x 6. x y + 36 Page 8

29 B) Perfect Square Trinomials: x bxy + cy Example: x 8xy + 16y A perfect square trinomial has main features: 1. The first term is a.. The term is a perfect square. The sign of the last term is always. 3. The term can be either positive or negative. It is always double the square root of the last term. Factoring a perfect square trinomial results in two Example: Factor: x - 8xy + 16y Check: Examples: Factor the following trinomials fully, if possible x +x. 5b 3-40b + 80b 3. The volume of a rectangular prism is represented by x 3-4x + 7x. What are possible dimensions of the prism? (V = LxWxH) Page 9

30 Assignment #7 Squares Factoring Difference of Squares and Perfect Factor the following DOS s and POS s: 1. n 5n 5. a t x h 6. 9c 16d 7. r s 8. 50g 7h 9. 9p 15r 10. s g 3h 1. y 1y x 6x z 1z a 5a b 4b Page 30

31 17. 16d 64e 18. 7m k 4k c c c a 5b. 3 s t 18s t 81st d lm 1lmn 3ln Page 31

32 ANSWER KEY: Assignment # Assignment # Assignment #3 1. GCF = 16. a) b) c) d) 3. a) GCF = b) GCF = 1 4. a) LCM = 60 b) LCM = LCM = 4 6. a) GCF = 6x b) GCF = 10cd c) GCF = bc d) GCF = 6xy z e) GCF = 5mn 7. a) GCF = 7 b) GCF = -5n c) GCF = 1 d) GCF = 4fg e) GCF = -15de f) GCF = 9j k Page 3

33 Assignment #4 a) 6(s + 5) b) 4(t + 7) c) 5(a 1) d) 4r(4r 3) e) 7x(y + y 7z) f) g) 5w(3w -1) h) i) 10xy(xy 5) j) g(g + ) k) 5xy(7x + 3xy + 1) l) ) Assignment #5 1. (x + 1)(x + 1). (3x 1)(x + ) 3. (3x )(x 3) 4. (x 3)(x + 4) 5. (x 5)(x + 1) 6. (3x 4)(x + 7) 7. (r + 7)(r + ) 8. (l + 3)(l + 4) 9. 3(w + )(w + 1) 10. (5b + 4b + 1) 11. (y + 3z)(y + z) 1. (4a + 1)(3a + 4) 13. (f 3)(f + 5) 14. (r + 11)(r 10) 15. 3(b + b 1) 16. (m 3n)(5m n) 17. (3g f)(3g f) 18. (3l + 7)(l + 3) Assignment #6 1. (y + 6)(y + ). (x + 7)(x + 3) 3. (a 10)(a 9) 4. (m 7n)(m + 6n) 5. (b + )(b + 17) 6. (g 6)(g 4) 7. (n 13)(n ) 8. (c 8)(c 7) 9. (s 5t)(s t) 10. Not Factorable Assignment #7 1. not factorable. (a 10)(a + 10) 3. (t 7)(t + 7) 4. Not factorable 5. (8 h)(8 + h) 6. (3c 4d)(3c + 4d) 7. (r s)(r + s) 8. (5g 6h)(5g + 6h) 9. 3(3p 5r ) 10. Not Factorable 11. 8(3g h)(3g + h) 1. (y + 6) 13. (x 3) 14. (z + 3) 15. Not Factorable 16. 4(6 b) Page 33

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