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Unit 2

Unit 2. Expressions. Section 1: Adding and Subtracting Like Terms. Like terms are those that have EXACTLY matching variables (order does not matter) You can add and subtract the coefficients to like terms by using the distributive property in reverse

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Unit 2

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  1. Unit 2 Expressions

  2. Section 1: Adding and Subtracting Like Terms • Like terms are those that have EXACTLY matching variables (order does not matter) • You can add and subtract the coefficients to like terms by using the distributive property in reverse • The Distributive Property: a(b + c) = ab + ac and (b + c)a = ab + ac • For example: 3x + 5x = (3 + 5)x = 8x • Ex1. Simplify: 3a + 2b – 8a + b • Ex2. Simplify

  3. Ex3. Simplify • If it helps, you can change subtraction signs to adding negative values • Ex4. Simplify 10x – 8y – 4x – (-2y) • If there is a negative or subtraction sign directly outside a set of parentheses containing either a sum or difference, distribute the sign to each term within the parentheses • Ex5. Simplify 10x – (5x + 8) + 12 – 3x • Ex6. Simplify (5n – 8p) – (9n – 5p) + 4p

  4. Opposite of a Sum Property: For all real numbers a and b, -(a + b) = -a + -b = -a – b • Opposite of Opposites Property (Op-op property): For a real number a, -(-a) = a • Opposite of a Difference Property: For all real numbers a and b, -(a – b) = -a + b • Ex7. Simplify • Ex8. Simplify • Sections from the book to read: 3-6 and 4-5

  5. Section 2: Simplifying Rational Expressions • A rational expression contains at least one fraction • You must have a common denominator in order to add or subtract fractions • Multiply the numerator and denominator of the fraction by the same number • Do this to both fractions so that the denominators are the same • Then add or subtract the numerators (combining like terms) and leave the denominator the same

  6. Simplify each rational expression • Ex1. • Ex2. • Ex3. • Ex4. • Sections of the book to read: 3-9, 4-5, and 5-9

  7. Section 3: Multiplying Monomials and Raising to a Power • When you are multiplying terms, add the exponents of the variables that are alike • Product of Powers Property: For all m and n, and all nonzero b, • Simplify • Ex1. • Ex2. • Ex3.

  8. When you raise a power to a power, multiply the exponents • Power of a Power Property: For all m and n, and all nonzero b, • Ex4. Simplify • If the exponent is directly outside of parentheses that contain a monomial, then you multiply every exponent inside of parentheses by the one outside • Power of a Product Property: For all nonzero a and b, and for all n,

  9. Simplify • Ex5. • Ex6. • Ex7. Solve for n. • Sections from the book to read: 2-5, 8-5, 8-8 and 8-9

  10. Section 4: Negative Exponents • A negative exponent does NOT make anything in the expression negative • Negative Exponent Property: For any nonzero b and all n, the reciprocal of • Only the power with the negative exponent is changed, it is moved to the other half of the fraction • Write with no negative exponents • Ex1. Ex2. Ex3. • Ex4. Write as a simple fraction

  11. Ex5. Write as a negative power of an integer • Zero Exponent Property: If g is any nonzero real number then, • Ex6. Write without negative exponents • Ex7. Simplify • Ex8. Simplify • Sections from the book to read: 8-2, 8-6, 8-9, and 12-7

  12. Section 5: Dividing Monomials and Raising to a Power • When dividing monomials, subtract the exponents of the matching variables • Quotient of Powers Property: For all m and n, and all nonzero b, • Write answers without negative exponents unless the directions allow it • Ex1. Simplify • Write as a simple fraction • Ex2. Ex3.

  13. Simplify. Write as a fraction with no negative exponents • Ex4. Ex5. • Power of a Quotient Property: For all nonzero a and b, and for all n, • Write as a simple fraction with no negative exponents • Ex6. Ex7. Ex8. • Sections from the book to read: 8-7, 8-8, 8-9

  14. Section 6: Multiplying and Dividing Rational Expressions • To multiply rational expressions, multiply the numerators together and the denominators together and be sure to simplify • You can simplify before you multiply or after • To divide rational expressions, flip the second expression and then multiply • Do NOT use mixed numbers with variables • Yes: or No:

  15. Simplify. Write the answer with no negative exponents. • Ex1. Ex2. • Ex3. Ex4. • Sections of the book to read: 2-3 and 2-5

  16. Section 7: Multiplying by Monomials and Binomials • Multiplying a monomial by a polynomial is using the distributive property • Write your answers in standard form • A subscript is NOT a mathematical process, it is just another name for a variable • i.e. x1 and x2 are two different variables • Multiply • Ex1. • Ex2.

  17. If you are multiplying a binomial by another binomial, FOIL will help make sure you don’t miss any terms • FOIL: First, Inner, Outer, Last • Multiply the First term in each binomial, then multiply the two Inner terms, then multiply the two Outer terms, then multiply the two Last terms, and finally combine like terms • Multiply • Ex3. (x + 4)(x + 6) Ex4. (m – 3)(m – 5) • Ex5. (n + 6)(n – 9) Ex6. (2a + 3)(a – 5) • Ex7. (3w² + 5)(2w² ─ 7) • Sections of the book to read: 3-7, 10-1, 10-3, and 10-5

  18. Section 8: Multiplying Polynomials • Use the Extended Distributive Property in order to multiply polynomials • Multiply every term in the first polynomial by every term in the second polynomial • See page 633 for a rectangular way to demonstrate this property • You can write the work vertically or horizontally (your choice) • Multiply • Ex1. • Ex2. • Sections of the book to read: 2-1 and 10-4

  19. Section 9: Special Binomial Products • Perfect Square Patterns: For all numbers a and b (a + b)² = a² + 2ab + b² and (a – b)² = a² - 2ab + b² • You can use this shortcut when multiplying • Square of a sum is a sum squared • i.e. (a + b)² • Square of a difference is a difference squared • i.e. (a – b)² • The result of a square of a sum and a square of a difference is called a perfect square trinomial • Expand • Ex1. (x – 5)² Ex2. (a + 7)² Ex3. (4m – 3)²

  20. If you multiply two binomials that are identical except one is addition and one is subtraction, the outer and inner terms will cancel out • The result is called the difference of squares • Difference of Two Squares Pattern: For all numbers a and b, (a + b)(a – b) = a² - b² • Expand • Ex4. (x + 5)(x – 5) Ex5. (3x – 2)(3x + 2) • You can use these patterns to do some basic arithmetic • Ex6. 43² Ex7. 81 · 79 • Section of the book to read: 10-6

  21. Section 10: Writing Expressions and Equations • Once you read the word “is,” that is where you put the equal sign • If the book uses the word “the quantity,” that is where you put the parentheses • Write an expression for each sentence • Ex1. The sum of 8 and the product of a number and 6 • Ex2. The quantity of a number plus seven will then be divided by 9 • Ex3. The difference of a 7 and a number • When given a table, look for a pattern to describe the situation

  22. Ex4. Write an equation based on the information • Ex5. Pencils sell for $0.24 each while notebooks sell for $0.72 each. Write an expression to describe how to find the total cost if you buy p pencils and n notebooks • Ex6. A parking lot charges $3 for the first hour and then $2 for every hour after that • A) If a car is in the lot for 6 hours, how much will the owner pay? • B) If a car is in the lot for h hours, how much will the owner pay? • Sections of the book to read: 1-7, 1-9, and 3-8

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