LOWELL CATHOLIC HIGH SCHOOL Summer Math ENTERING ALGEBRA II & TRIGONOMETRY Simplifying Expressions Order of Operations Parenthesis/grouping symbols Exponents Multiplication Division Addition Subtraction 1. Simplify the expression within each grouping symbol, working outward from the innermost grouping. 2. simplify powers 3. Perform multiplication and divisions in order from left to right. 4. Perform additions and subtractions on order from let to right. Example 25 6 3 25 6 3 10 6 3 10 2 8 Simplify. 1) 11(3 5) 2 4) 3 (8 45) 2) 36 9 4 5) 38 45 3) (42 6) 2 3 Simplifying Expressions Substitution Principle An expression may be replaced by another expression that has the same value. Example Evaluate if x = 4 and y = 3 into the expression 2xy 4x2 Solution: 2xy 4x 2 243 442 243 416 24 64 40 Evaluate 6) x y 3z if x 7, y 8, and 9) 3y2 y 5 if x = 3, y = 2, and z=5 z 0 3x y 10) 3(x 1) if x = 3 7) if x 5 and y 3. 3x y 8) (yz x)3 if x = 3, y = 2, and z = 5 Inequalities Symbols and Sentences Inequality Symbol , , , , are all inequality symbols. is read “less than or equal to” and is read “greater than or equal to”. Inequality An inequality is a sentence formed by placing an inequality symbol between two expressions. Use one of the inequality symbols to make a true statement. 11) 5 _____ 4 14) 51 ____ 5 1 12) 2 2 _____ 2 2 3 2 4 2 ____ 15) 3 2 4 2 13) (18 6) 3 ____ 18 (6 3) Words into Algebra Algebra is often used as a problem-solving tool. In order use this tool we often must change word phrases into algebraic expressions. Example Word Phrase Algebraic Expression A number increased by 10 n + 10 15 more than a number n + 15 The sum of a number and 67 n + 67 A number decreased by 22 n – 22 * 82 less than a number n – 82 * The difference between a number and 182 n - 182 The product of a number and 43 43n A number times 5.5 5.5n The quotient of 10 and a number A number divided by 5 * Notice the order of the terms of the expression. 22 – n is not the same as n – 22. Translate the following word phrases into algebraic expressions. Use n for the variable in each problem. 16) A number decreased by 2 20) The quotient of 5 times a number and 2 times a number 17) 5.2 times a number 21) 2 times the sum of a number and 5 18) The sum of a number and 2 times the number 22) One more than the square of a 19) 15 less than a number number Addition of Fractions Example 1: Fractions with a common denominator. 3 1 5 8 8 8 Find the sum 3 1 5 (3 1 5) 9 1 or 1 8 8 8 8 8 8 Example 2: Fractions without a common denominator. 3 5 1 4 6 2 Find the sum 3 5 1 (3 3) (5 2) (1 6) (9 10 6) 25 1 or 2 4 6 2 (4 3) (6 2) (2 6) 12 12 12 5 1 2 6 3 2 1 23) 26) 7 7 7 7 5 15 3 1 5 2 11 5 7 24) 27) 3 6 3 18 18 18 1 1 1 25) 2 4 3 Subtraction of Fractions Example 1: Fractions with a common denominator. 3 1 5 8 8 8 Find the difference 3 1 5 (3 1 5) 3 8 8 8 8 8 Example 2: Fractions without a common denominator. 5 3 6 4 Find the difference 5 3 (5 2) (33) (10 9) 1 6 4 (6 2) (43) 12 12 5 1 5 2 28) 31) 6 6 6 15 5 1 7 2 29) 32) 8 6 8 3 13 11 30) 29 29 Multiplication of Fractions Example 1: 2 4 3 5 Find the product 2 4 Product of numerators 2 4 8 3 5 Product of deno min ators 35 15 3 7 9 5 33) 36) 4 8 10 3 2 3 7 34) 660 37) 3 11 2 12 21 35) 35 30 Division of Fractions Example 1: 2 4 3 5 Find the quotient 2 4 2 5 2 5 10 5 3 5 3 4 3 4 12 6 3 1 9 1 38) 41) 4 2 16 4 7 1 15 3 39) 42) 8 3 8 4 8 4 40) 5 3 Sums and Differences of Positive and Negative Numbers Rules for adding and subtracting any real numbers: 1) If two numbers have the same sign, add their absolute values and use their common sign (+ or -). 2) If two numbers have opposite signs, subtract the lesser number from the larger number and use the sign of the larger number. Example Expression Answer Explanation Both signs are the same so add the numbers and 4 6 10 take the common positive sign Both signs are different. Subtract 4 from 6 and 4 6 2 take the sign of the larger number (positive 6) Both signs are the same so add the numbers and 4 (6) 10 take the common sign ( Both signs are different. Subtract 4 from 6 and 4 (6) 2 take the sign of the larger number (negative because 6 is larger than 4) Simplify. 43) 32 53 44) 68 (42) (35) 61 45) 87 16 (22) 61 46) 10.2 17.6 47) 15 40 Sums and Differences of Positive and Negative Numbers Definition of subtraction Not every algebraic operation is an addition operation. To help make the subtraction operation easier, we can change it to an addition operation. To make this change, we use the definition of subtraction a – b = a + (-b) Another way to state this is to make the subtraction operation an addition operation and change the sign of the term following the operation. Example 25 50 25 50 75 Find each difference. 48. 5 12 49. 7 (14) 50. 7 (7) 51. 7 7 52. 7 (7) Combining Like Terms Like (or similar) terms Terms which have the same variables with the same exponents Use the distributive property if necessary to simplify before combining like terms. Distributive Property Property of the Opposite of a Sum a(b + c) = ab + ac (a b) (a) (b) a(b – c) = ab – ac Example: 4y 2(6y 5) 4y 12y 10 16y 10 Simplify. 53. 6y 3z 3y 56. 13a 13b 13c 15a 54. 6y (3y 10) 57. 5x 6x (3x 5) 55. 3y 4x 2(4x 5y) 58. 4(3 y) 2(1 y) Products of Real Numbers Rules of Multiplication 1. The product of two positive numbers or two negative numbers is a positive number. 2. The product of a positive number and a negative number is a negative number. Simplify. 1 62) 7k 4(3k 6) 59) ( )(4r)(s) 2 63) 3(p 5) 7p 60) 5(2)(7)(3) 64) (a)(2b)(3c) 61) 17(13) 17(7) Quotients of Real Numbers Rules for Division 1) The quotient of two positive numbers or two negative numbers is a positive number. 2) The quotient of two numbers when one is positive and the other negative is a negative number. For a real numbers a and b and nonzero real numbers c, a b a b a b a b and c c c c c c Simplify. 65) 21 7 24 6x 2 68) 2 (3)(4)(2) 66) (6)(2) 36x 2 24x 6 69) 6 32 52 67) 3 (5) 9x 2 27 70) 3 Solving Equations Example: 7x 13 50 7x 13 13 50 13 7x 63 7x 63 7 7 x 9 28 x 8x 28 x x 8x x You want to get the 28 7x variables on one side and numbers 28 7x on the other. 7 7 4 x or x 4 Solve. 71) 9y 8 80 76) 18 3y 57 72) 3x 5 5 2x 77) 5(m 4) 8(m 2) 73) 3b 7b 35 15 2b m 4 78) 4 6 74) 3x 10 5(x 4) 75) 4x 83 1 Working with Polynomials Adding Polynomials Definition Example Constant A number 1 2, , 0 2 Monomial a constant, a variable, or a product 1 3, 5x, x 2 y3, x5 of a constant and one or more 4 variables Coefficient The constant factor of a monomial The coefficient of 3m2 is 3 Polynomial A monomial or a sum of 3x2 5x 7 monomials. The monomials in a The terms of the polynomial are called the terms of polynomial are 3x 2 , 5x , the polynomial. 7 To add two or more polynomials, write their sum and then simplify by combining similar terms. Example: Simplify by adding the following polynomials. Add 2x2 3x 5 and x3 5x2 2x 5. 79) 5m 4 (2m 3) (2x2 3x 5) (x3 5x2 2x 5) x3 (2x2 5x2 ) (3x 2x) (5 5) x3 3x2 x 80) 2n2 n 5 (n2 1) 81) 4a2 3ab b2 (b2 2ab) 82) w3 w 2 w 1 (1 w w 2 w3 ) 83) 2(4m2 3) 7(m2 2) 1 84) 3x2 2xy 4y2 (2x2 3y2 ) Working with Polynomials Subtracting Polynomials To subtract one polynomial from another, add the opposite of each term of the polynomial you are subtracting. Example: Subtract 2x2 3x 5 from x3 5x2 2x 5 x3 5x2 2x 5 (2x2 3x 5) x3 5x2 2x 5 (2x2 3x 5) x3 (5x2 2x2 ) (2x 3x) (5 (5)) Subtract.