How could you express algebraically, the total amount of money he earned for the three days?

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1 UNIT 4 POLYNOMIALS

2 Math 11 Unit 4 Introduction p. 1 of 1 A. Algebraic Skills Unit 4 Polynomials Introduction Problem: Derrek has a part time job changing tires. He gets paid the same amount for each tire he changes. Here is a list of the number of tires he changed for three days: Monday: 6 tires Tuesday: 8 tires Wednesday: 10 tires How could you express algebraically, the total amount of money he earned for the three days? Let x = the amount of money earned per tire Solution: 6x + 8x + 10x = 4x If Derrek earns $5.00 per tire, how much does he make? Solution: 4(5) = $10. Derrek earns $10 over the 3 days. A polynomial is an expression that uses numbers and variables (letters) to simplify a math problem. The terms in a polynomial can be added, subtracted, multiplied, and divided in order to simplify the expression.

3 Math 11 Unit 4 Objectives p. 1 of 1 A. Algebraic Skills Unit 4 Polynomials Objectives To become familiar with polynomial terminology. To add and subtract polynomials. To multiply a monomial by a monomial. To multiply a polynomial by a monomial. To divide a polynomial by a monomial.

4 Math 11 Unit 4 Outline p. 1 of 1 A. Algebraic Skills Unit 4 Polynomials 1. Polynomial Terminology Exercise 1. Adding and Subtracting Polynomials Exercise 3. Multiplication of Monomials by Monomials Exercise 3 4. Multiplication of Polynomials by Monomials Exercise 4 5. Division of Polynomials by Monomials Exercise 5 Unit 4 Review Unit 4 Exam

5 Math 11 Unit 4 Lesson 1 p. 1 of Math 11 Unit 4 Lesson 1 SWBAT become familiar with polynomial terminology 1. Polynomials Term: - basic element of all algebraic expressions - terms are separated by a + or sign - has components: 1. Numerical Coefficient (number). Literal Coefficient (variable/letter and exponent) e.g. 17x 3 Term Numerical Coefficient Literal Coefficient Constant: Polynomial: Monomial: Binomial: Trinomial: Degree: a term with no variables; a number e.g. -5 an algebraic expression containing one or more terms. one term e.g. 7x 3 two terms e.g. 8x + y three terms e.g. 10x + 6xy + 3y is the greatest exponent total of any term in a polynomial Standard Form: when a polynomial is written in descending (large to small) order of degrees (exponents) from left to right

6 Math 11 Unit 4 Lesson 1 p. of State the degree for the following polynomials and what type they are: Ex. 1. a b 6 c 3 Degree: = 11 Type: monomial Ex.. 7k 3 3r + 5kr 3 Degree: choose highest = 3 Type: trinomial Ex. 3. 9x 4 y xy Degree: 11 Type: binomial Arrange the terms in descending order of degrees. Ex. 4. 5b 5b 4 9b 7 b 3 = -5b 4 b 3 9b + 5b 7 Ex. 5. 3x y + y 7 x 3 in terms of x = -x 3 + 3x y + y - 7 Exercise 1

7 Math 11 Unit 4 Exercise 1 p. 1 of Math 11 Unit 4 Exercise 1 Fill in the blanks using the following words. Some words may be used twice. exponent one monomial ascending letter two binomial descending term three trinomial number four polynomial 1. A is a basic element of all algebraic expressions.. A degree is the greatest of any term in the polynomial. 3. A has two terms. 4. A numerical coefficient is the same as a. 5. A polynomial is written in standard form when the terms are in order of degrees from left to right. 6. There is/are term(s) in the polynomial 3xy + 7x 8yz. 7. A literal coefficient is the same as a. 8. The degree of the polynomial abcd is. 9. A has one term. 10. A term has components. 11. There is/are term(s) in the polynomial 9a + 5b. 1. A trinomial has terms. 13. A is an algebraic expression containing one or more terms. 14. There is/are term(s) in the polynomial 3xy. 15. The coefficient in the expression, 4st, is the.

8 Math 11 Unit 4 Exercise 1 p. of State the degree for the following polynomials and what type they are: 16. 8a + 5a a 6-9a 3 b. 15a 9 + 7a gh 0. 4g 4 h 5 j + g h 3 j + 3g 6 j 3. 5r t + 8r 3 t u 1. 7y + 8y a 5 b + a 3 b gh 3 j 10 Arrange in descending order of degrees: 6. 5a + 4a 4 a y 5 + 3y + 1y d 3 + d 3 +d 4 d 3. 9t 13 9t t t 9t t a 5 + 9a 4a 1 + a xy 4 + 5x 3 y x in terms of x 34. 6xy + 1x y 3x 6 + 7x 8 in terms of x 30. a 5 b ab 7 + a 4 b 6 + a 3 b 8 in terms of a f 3 + f 5f 10 +6f 4 9f + f 7 7f 1

9 Math 11 Unit 4 Lesson p. 1 of Math 11 Unit 4 Lesson SWBAT add and subtract polynomials.. Adding and Subtracting Polynomials You can only add and subtract like terms Like Terms: same literal coefficient (same variable and exponent) Examples of like terms: o 7x, -8x, 13x like term is x o 1abc, -abc, 3abc like term is abc o 5r t, 10r t, -r t like term is r t Add or subtract the coefficient (number), but keep the variable (letter) the same. Add or subtract like terms: Ex a + 7a = a Ex.. 9w 3w + w = 8w Ex. 3. 7x (-6y) 3x + y = 7x 3x + 6y + y {Group like terms} = 4x + 8y Ex. 4. 4ab + 7ab 1 + ab 3ab + 8 = 4ab + 7ab + ab 3ab = 10ab + 7

10 Math 11 Unit 4 Lesson p. of * When subtracting a polynomial in brackets, remember to subtract all terms inside the brackets. Ex. 5. (6b 3c) - (-8b + 9c) = 6b 3c + 8b - 9c = 6b + 8b 3c - 9c = 14b - 1c Ex. 6. (4x + 9) (x 5) = 4x + 9 x + 5 = 4x x = x + 14 Exercise

11 Math 11 Unit 4 Exercise p. 1 of Math 11 Unit 4 Exercise Add or subtract like terms. 1. 5a + 3a 10. 6b + (-3b) 7b. 5k + 8k 11. 8h (-7h) 3. 8n 1n 1. 11x 4y + 9y 16x 4. 4w 7w 13. 7p 8r 5p 3r 5. 8s 13s 14. 9b + 9a + b a + 7b 6. 5y 9y y 15. 6h h h 7. 4t 5t + 9t mn + 6pq + mn 5pq 8. -5n + 9n + 16n + 9n rs (-wx) 6rs + wx 9. a (-6a) 18. 4y 10c + 6y c +(-4y) + c

12 Math 11 Unit 4 Exercise p. of 19. (1a 3c) + (3a + 4c) 5. (7r 3p) (-6r + 9p) 0. (7y ) + (y 17) 6. (x + 9) + (-x 7) 1. (11k 4m) (k 15m) 7. (6x 17) (-7x ). (-7y ) + (y + 17) 8. (4a + b) (6a b) 3. (11k 4m) + (-k 15m) 9. (-y + 18) (-y + 17) 4. (4x + 8y) + (x 10y) 30. (7x + y) (7x y)

13 Math 11 Unit 4 Lesson 3 p. 1 of 1 Math 11 Unit 4 Lesson 3 SWBAT multiply a monomial by a monomial. 3. Multiplication of Monomials by Monomials Steps: 1. Multiply the coefficients (the numbers). Write down the common bases (letters) 3. Add the exponents of the common bases Multiply: Ex. 1. 5(9p 3 ) = -45p 3 Ex.. 3x 3 7x 6 = (3 7) (x 3 x 6 ) = 1x 9 Ex. 3. (3m 3 n 4 )(-7m 5 n) = -1m 8 n 5 Ex s t 3 x 3st 4 = 30s 3 t 7 Ex. 5. 3a bc 5b 3 c -a 3 b 4 = 30a 5 b 8 c 3 Exercise 3

14 Math 11 Unit 4 Exercise 3 p. 1 of Math 11 Unit 4 Exercise 3 Multiply (3y). -4(-8a) 3. 3c 8c 4. -5x 4x 5. 8w 9w 6. (-a)(-1a 5 ) 7. 6t 3 8t p 5p s 5 6s m 11m c 3 d c d a 6 b 5a 8 b (-3x y 4 )(8xy 3 ) 14. 7h 4 k 8hk 3

15 Math 11 Unit 4 Exercise 3 p. of k 6 m -k 3 m 16. (4xy 5 )(-7xy) 17. 3w 4 x y 9wx 3 y (-6m p 3 r )(-6m pr ) 19. y 5 6y 3y 7 0. a 7 a 3 a 1. (3y)(1y )(y ). (-7c 3 )(-c)(-5c 4 ) 3. (a)(3b 3 )(-8ab 5 ) 4. (-5)(a y 5 )(1a 6 y) 5. (-9ax)(-3a x 3 )(-4x) 6. (4r 4 )(-7p 7 r 3 )(-pr ) 7. (m 5 r )(3m)(-r p 7 ) 8. (-14a )(-b 3 c)(-a 3 bc ) 9. (4pr )(5p r 3 ) 30. (-xy )(-x y)

16 Math 11 Unit 4 Lesson 4 p. 1 of 1 Math 11 Unit 4 Lesson 4 SWBAT multiply a polynomial by a monomial. 4. Multiplication of Polynomials by Monomials We use the distributive property to multiply. Multiply the monomial on the outside of the brackets by each term inside the brackets. Multiply: Ex. 1. 5(x 3 + 3x) = (5 x 3 ) + (5 3x) = 10x x Ex.. -r(7r 3 10r 6 ) = (-r 7r 3 ) (-r 10r 6 ) = -14r 4 + 0r 7 Ex. 3. 3x (x + 4x ) = 6x 4 + 1x 3-6x Ex. 4. 3s t (10s t 3 + 3st 4 st) = 30s 4 t 5 + 9s 3 t 6-6s 3 t 3 Exercise 4

17 Math 11 Unit 4 Exercise 4 p. 1 of Math 11 Unit 4 Exercise 4 Multiply. 1. 7(y 5). f(f + 3f) 3. 8z(z 3z) 4. 6d(3d 3 + d ) 5. n (10n 7) 6. v 4 (8v + 5v) 7. xy(9x y 3 x) 8. cd 3 (3cd + 4c 4 ) 9. 11ab(3a 3 b + 11) 10. 6(3y 3 9y 5y) 11. n(5n 4 n + 4) 1. 3p(4p 3 + 5p - 1) 13. t 3 (-3t 3 + t - 7)

18 Math 11 Unit 4 Exercise 4 p. of 14. 5w 4 (w 4 w + w) 15. cdf(c + cd + cdf) 16. p r 3 (pr + 4p 3r) 17. 9ab(3a b + ab - 1) (t 3 t 7t + 5) 19. 6k(k 3 k + 7k - 1) 0. z 3 (z 3 + 5z + 3z + 4) 1. -3h(4h 4 + h 3 6h 6h). -xy(3xz + y - y 3yz) 3. a 3 b(ab 4 + a b b + b) 4. -k (-9k 5-6k 3-1) 5. -ab (-1a 4 b 3 + 3abc b + a b)

19 Math 11 Unit 4 Lesson 5 p. 1 of 1 Math 11 Unit 4 Lesson 5 SWBAT divide a polynomial by a monomial. 5. Division of Polynomials by Monomials Divide each term of the polynomial by the monomial. Remember to subtract exponents of like bases. Divide: Ex. 1. 1v 4v = 3v Ex a 5a = -3a 4 Ex c 7 d 3 8c 5 d 3 = 6c Ex x + 4x 6x 3 5 = 18x 6x 3 4x + 6x 5 6 1f 18 f Ex f = 3x + 4x 4 4 1f 3 f 18 f 3 f 6 4 = = 7f 4 6f Exercise 5

20 Math 11 Unit 4 Exercise 5 p. 1 of Divide. Math 11 Unit 4 Exercise m 5m 4w rp 3 rp 11. 5n 4 5n 1. 4p 3 (-1p ) 3. 7a 3 8a t 10 t 4 100h h w 1 15w c d cd 5. 36w 8 1w x a 6. 9a m n m 7. 3 n m w h k 4hk a m 8 r 10 10mr a 11b 11

21 Math 11 Unit 4 Exercise 5 p. of 1. 4 d y 3 4y d 4d 5 + 8d 6 8x + 1x x n 7n 7n 5 4a + 16a 6. 4a f 7 f f g 64g g w x 14w x 7w x 6 5 3x y 40x y 30. 8xy 4

22 Math 11 Unit 4 Review p. 1 of 4 Name: Date: Math 11 Unit 4 Review What is the type and degree of the following polynomials: 1. 8d 3 + 7d. 10t 5 u a b 3 c 4. -8x 4 y 5 + 9y 7 3xy x 6 y 3 z 6. 4n m 3 4nm 7. -7x 1x y + 4y 3 Write the following polynomials in descending order: 8. 9x 3 + 4x 7 9x f + 1f 8-3f 7 + f 14f d d 6 + 9d

23 Math 11 Unit 4 Review p. of 4 Add or subtract the following: 11. 4w + 7w 1. 11uw + (-1uw) + 4uw y + 5y + 17y + (-9y) 14. 7x x p 7 p 16. 3x 5 9x m 6n + p m 5n + 3p 18. 4m 11m 7 + 7m p 7p p 4p 0. (n + 4) (n 5) 1. (3x 5y) (7x + y)

24 Math 11 Unit 4 Review p. 3 of 4 Multiply the following:. 6m 6 3m 4 3. (5z )(4z 3 ) 4. 4m 11m k 6 m -k 3 m 6. (3x)(-x 4 )(-x ) 7. (-7c 3 )(-c)(-5c 4 ) 8. 8(x - 4) 9. -5(r + 3) 30. m(5m + 3) d (d + 3) 3. f (3f 3 + 5) 33. 3j 3 (4j 3 7j ) 34. p(4p 3 + 5p 1)

25 Math 11 Unit 4 Review p. 4 of 4 Divide the following: 8 36w w 6 4 p p h 5 i 8 11h 5 i y y 6y 3 + 6y z 40z 8z

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