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### 4 3polynomial expressions

1. 1. Polynomial Expressions
2. 2. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. Polynomial Expressions
3. 3. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Polynomial Expressions
4. 4. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Polynomial Expressions
5. 5. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous.
6. 6. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous.
7. 7. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous.
8. 8. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
9. 9. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. The simplest form of expression are #xN, where N is a non- negative integer and # is a number, is called a monomial (one-term). Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
10. 10. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. The simplest form of expression are #xN, where N is a non- negative integer and # is a number, is called a monomial (one-term). For example –1, 2x, 3x2, and –4x3 are monomials. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
11. 11. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. The simplest form of expression are #xN, where N is a non- negative integer and # is a number, is called a monomial (one-term). For example –1, 2x, 3x2, and –4x3 are monomials. If N = 0 we’ve the constants, N = 1, the linear monomials #x. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
12. 12. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Example A. Evaluate the monomials if y = –4 a. 3y2 The simplest form of expression are #xN, where N is a non- negative integer and # is a number, is called a monomial (one-term). For example –1, 2x, 3x2, and –4x3 are monomials. If N = 0 we’ve the constants, N = 1, the linear monomials #x. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
13. 13. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Example A. Evaluate the monomials if y = –4 a. 3y2 3y2  3(–4)2 The simplest form of expression are #xN, where N is a non- negative integer and # is a number, is called a monomial (one-term). For example –1, 2x, 3x2, and –4x3 are monomials. If N = 0 we’ve the constants, N = 1, the linear monomials #x. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
14. 14. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Example A. Evaluate the monomials if y = –4 a. 3y2 3y2  3(–4)2 = 3(16) = 48 The simplest form of expression are #xN, where N is a non- negative integer and # is a number, is called a monomial (one-term). For example –1, 2x, 3x2, and –4x3 are monomials. If N = 0 we’ve the constants, N = 1, the linear monomials #x. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
15. 15. b. –3y2 (y = –4) Polynomial Expressions
16. 16. b. –3y2 (y = –4) –3y2  –3(–4)2 Polynomial Expressions
17. 17. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. Polynomial Expressions
18. 18. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 Polynomial Expressions
19. 19. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 Polynomial Expressions
20. 20. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) Polynomial Expressions
21. 21. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 Polynomial Expressions
22. 22. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 Polynomial Expressions Polynomial Expressions
23. 23. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 The sum of monomials are called polynomials (many-terms). Polynomial Expressions Polynomial Expressions
24. 24. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 The sum of monomials are called polynomials (many-terms). These are expressions of the form, arranged in the order of powers of the x: #xN ± #xN-1 ± … ± #x1 ± # where the #’s are numbers. Polynomial Expressions Polynomial Expressions
25. 25. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 The sum of monomials are called polynomials (many-terms). These are expressions of the form, arranged in the order of powers of the x: #xN ± #xN-1 ± … ± #x1 ± # where the #’s are numbers. The highest exponent N is the degree of the polynomial. Polynomial Expressions Polynomial Expressions
26. 26. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 The sum of monomials are called polynomials (many-terms). These are expressions of the form, arranged in the order of powers of the x: #xN ± #xN-1 ± … ± #x1 ± # where the #’s are numbers. The highest exponent N is the degree of the polynomial. For example, 4x – 7 is 1st degree (linear) Polynomial Expressions Polynomial Expressions
27. 27. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 The sum of monomials are called polynomials (many-terms). These are expressions of the form, arranged in the order of powers of the x: #xN ± #xN-1 ± … ± #x1 ± # where the #’s are numbers. The highest exponent N is the degree of the polynomial. For example, 4x – 7 is 1st degree (linear) and the degree of 1 – 3x2 – πx40 is 40. Polynomial Expressions Polynomial Expressions
28. 28. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 The sum of monomials are called polynomials (many-terms). These are expressions of the form, arranged in the order of powers of the x: #xN ± #xN-1 ± … ± #x1 ± # where the #’s are numbers. The highest exponent N is the degree of the polynomial. For example, 4x – 7 is 1st degree (linear) and the degree of 1 – 3x2 – πx40 is 40. x 1 is not a polynomial.The expression Polynomial Expressions Polynomial Expressions
29. 29. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. Polynomial Expressions
30. 30. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. Polynomial Expressions
31. 31. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Polynomial Expressions
32. 32. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, Polynomial Expressions
33. 33. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3 Polynomial Expressions
34. 34. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3 = 4(9) – 3(–27) Polynomial Expressions
35. 35. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3 = 4(9) – 3(–27) = 36 + 81 = 117 Polynomial Expressions
36. 36. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3 = 4(9) – 3(–27) = 36 + 81 = 117 Given a polynomial, each monomial is called a term. Polynomial Expressions
37. 37. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3 = 4(9) – 3(–27) = 36 + 81 = 117 Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms Polynomial Expressions
38. 38. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3 = 4(9) – 3(–27) = 36 + 81 = 117 Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms Therefore the polynomial –3x2 – 4x + 7 has 3 terms, –3x2 , –4x and + 7. Polynomial Expressions
39. 39. Each term is addressed by the variable part. Polynomial Expressions
40. 40. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, Polynomial Expressions
41. 41. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, Polynomial Expressions
42. 42. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. Polynomial Expressions
43. 43. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. Polynomial Expressions
44. 44. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Polynomial Expressions
45. 45. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Operations with Polynomials Polynomial Expressions
46. 46. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Operations with Polynomials Polynomial Expressions
47. 47. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. Operations with Polynomials Polynomial Expressions
48. 48. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x Operations with Polynomials Polynomial Expressions
49. 49. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Operations with Polynomials Polynomial Expressions
50. 50. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. Operations with Polynomials Polynomial Expressions
51. 51. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Operations with Polynomials Polynomial Expressions
52. 52. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Note that we write 1xN as xN , –1xN as –xN. Operations with Polynomials Polynomial Expressions
53. 53. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it with the coefficient. Operations with Polynomials Polynomial Expressions
54. 54. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x Operations with Polynomials Polynomial Expressions
55. 55. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, Operations with Polynomials Polynomial Expressions
56. 56. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x. Operations with Polynomials Polynomial Expressions
57. 57. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x. Operations with Polynomials When multiplying a number with a polynomial, we may expand using the distributive law: A(B ± C) = AB ± AC. Polynomial Expressions
58. 58. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) Polynomial Expressions
59. 59. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x Polynomial Expressions
60. 60. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 Polynomial Expressions
61. 61. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) Polynomial Expressions
62. 62. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 Polynomial Expressions
63. 63. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Expressions
64. 64. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. Polynomial Expressions
65. 65. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers Polynomial Expressions
66. 66. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 Polynomial Expressions
67. 67. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Polynomial Expressions
68. 68. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. Polynomial Expressions
69. 69. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. For example 3x2y3 + 5x2y3 = 8x2y3 Polynomial Expressions
70. 70. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. Polynomial Expressions
71. 71. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y. Polynomial Expressions
72. 72. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y. If only one number is given, the result is a formula. Polynomial Expressions
73. 73. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y. If only one number is given, the result is a formula. If both numbers are given, then we get a numerical output. Polynomial Expressions
74. 74. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y. If only one number is given, the result is a formula. If both numbers are given, then we get a numerical output. We may do this for x, y and z or even more variables. Polynomial Expressions
75. 75. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 Polynomial Expressions
76. 76. = 6xy – 8x2y + 2xy – 3xy2 Polynomial Expressions Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2
77. 77. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 Polynomial Expressions
78. 78. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Polynomial Expressions
79. 79. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 Polynomial Expressions
80. 80. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 Polynomial Expressions
81. 81. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 Polynomial Expressions
82. 82. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 Polynomial Expressions
83. 83. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 We may put x = 2, y = 3 into the formula and do everything all over Polynomial Expressions
84. 84. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 We may put x = 2, y = 3 into the formula and do everything all over again or we may plug into y = 3 into part b which is easier. Polynomial Expressions
85. 85. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 We may put x = 2, y = 3 into the formula and do everything all over again or we may plug into y = 3 into part b which is easier. We will do the easy way. Polynomial Expressions
86. 86. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 We may put x = 2, y = 3 into the formula and do everything all over again or we may plug into y = 3 into part b which is easier. We will do the easy way. Input y = 3 into –16y – 6y2 Polynomial Expressions
87. 87. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 We may put x = 2, y = 3 into the formula and do everything all over again or we may plug into y = 3 into part b which is easier. We will do the easy way. –16(3) – 6(3)2 Input y = 3 into –16y – 6y2 we get Polynomial Expressions
88. 88. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 We may put x = 2, y = 3 into the formula and do everything all over again or we may plug into y = 3 into part b which is easier. We will do the easy way. –16(3) – 6(3)2 Input y = 3 into –16y – 6y2 we get = –48 – 54 = –102 Polynomial Expressions
89. 89. Ex. A. Evaluate each monomials with the given values. 3. 2x2 with x = 1 and x = –1 4. –2x2 with x = 1 and x = –1 5. 5y3 with y = 2 and y = –2 6. –5y3 with y = 2 and y = –2 1. 2x with x = 1 and x = –1 2. –2x with x = 1 and x = –1 7. 5z4 with z = 2 and z = –2 8. –5y4 with z = 2 and z = –2 B. Evaluate each monomials with the given values. 9. 2x2 – 3x + 2 with x = 1 and x = –1 10. –2x2 + 4x – 1 with x = 2 and x = –2 11. 3x2 – x – 2 with x = 3 and x = –3 12. –3x2 – x + 2 with x = 3 and x = –3 13. –2x3 – x2 + 4 with x = 2 and x = –2 14. –2x3 – 5x2 – 5 with x = 3 and x = –3 C. Expand and simplify. 15. 5(2x – 4) + 3(4 – 5x) 16. 5(2x – 4) – 3(4 – 5x) 17. –2(3x – 8) + 3(4 – 9x) 18. –2(3x – 8) – 3(4 – 9x) 19. 7(–2x – 7) – 3(4 – 3x) 20. –5(–2 – 8x) + 7(–2 – 11x) Polynomial Expressions
90. 90. 21. x2 – 3x + 5 + 2(–x2 – 4x – 6) 22. x2 – 3x + 5 – 2(–x2 – 4x – 6) 23. 2(x2 – 3x + 5) + 5(–x2 – 4x – 6) 24. 2(x2 – 3x + 5) – 5(–x2 – 4x – 6) 25. –2(3x2 – 2x + 5) + 5(–4x2 – 4x – 3) 26. –2(3x2 – 2x + 5) – 5(–4x2 – 4x – 3) 27. 4(3x3 – 5x2) – 9(6x2 – 7x) – 5(– 8x – 2) 29. Simplify 2(3xy – xy2) – 2(2xy – xy2) then evaluated it with x = –1, afterwards evaluate it at (–1, 2) for (x, y) 30. Simplify x2 – 2(3xy – x2) – 2(y2 – xy) then evaluated it with y = –2, afterwards evaluate it at (–1, –2) for (x, y) 31. Simplify x2 – 2(3xy – z2) – 2(z2 – x2) then evaluated it with x = –1, y = – 2 and z = 3. Polynomial Expressions 28. –6(7x2 + 5x – 9) – 7(–3x2 – 2x – 7)