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- 1. Polynomial Expressions
- 2. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. Polynomial Expressions
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. b. –3y2 (y = –4) Polynomial Expressions
- 16. b. –3y2 (y = –4) –3y2 –3(–4)2 Polynomial Expressions
- 17. b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48. Polynomial Expressions
- 18. b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48. c. –3y3 Polynomial Expressions
- 19. b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48. c. –3y3 –3y3 – 3(–4)3 Polynomial Expressions
- 20. b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48. c. –3y3 –3y3 – 3(–4)3 = – 3(–64) Polynomial Expressions
- 21. b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48. c. –3y3 –3y3 – 3(–4)3 = – 3(–64) = 192 Polynomial Expressions
- 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. 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. 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. 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. 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. 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. 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. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. Polynomial Expressions
- 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. 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. 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. 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. 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. 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. 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. 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. 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. Each term is addressed by the variable part. Polynomial Expressions
- 40. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, Polynomial Expressions
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) Polynomial Expressions
- 59. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x Polynomial Expressions
- 60. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 Polynomial Expressions
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 Polynomial Expressions
- 76. = 6xy – 8x2y + 2xy – 3xy2 Polynomial Expressions Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2
- 77. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 Polynomial Expressions
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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)

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