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.
x3 7x2 5x 10 85) 4a4 5 (2a4 1) 88) 5x3 3x 7 (4x2 5)
86) (8y3 3y 9) (8y2 2y 7) 89) y4 2y2 y 1(y4 3y3 3y 1)
87) 6x4 5x (5x5 2x4 3x2 2x 1) 90) 9m3 (3m2 2m 1)
Multiplying Using Exponents
Law of Exponents when multiplying a common base:
am an amn
Example:
c5 c 3 c53 c8 35 34 354 39 (mn4 )(m5n6 ) (mm5 )(n4 n6 )
= (m15 )(n46 )
= m6n10 Simplify. *m is the same as m1
91) x3 x10
92) mmm
93) (5)4 (5)3
94) (xyz )(x3y2z4 )
95) 99 95 9
96) (3a4b5 )(3a2b3 ) Multiplying Using Exponents
Law of Exponents when raising a product to a power.
m m m (ab) a b
Example: (xy )3 x3y3 (3ab)2 32 a2 b2 2 2 = 9a b
Law of Exponents when simplifying a power to a power.
(am )n amn
Example: (x3 )4 x34 x12 (2x4 y3 )2 22 (x42 )(y32 )
8 6 = 4x y
Simplify.
97) (4a3b2 )2
98) (s2t)3 (st3 )2
99) (3pq4r 2 )3
100) p2q3 (p2 4q)
101) x4 xk4
102) (x 4 )3 Division Using Exponents
Law of Exponents when dividing two monomials m a mn If m n, then a a n
m a 1 If n m, then n nm a a
m a a m m b b
Examples:
8 2 3 3 3 33 9 9 285 23 5x 5 5 t t t t 5 2 7 73 4 13 3 x x x 3 3 3 27
Simplify. Assume that no denominator equals 0.
3 3 18x 3r 103) 106) 6x s 2
3 5t 2 2 2 104) 4r s 5 107) 15t 2 4r 2 s
a 2b3c 105) 3 3 3 2 2hk a bc 108) h 2 k 2
Multiplying Polynomials
To multiply two polynomials, you use the distributive property: Multiply each term of one polynomial by each of the other and add the resulting monomials.
Example:
Multiply 2x 3x2 4x 5
2x 3x2 4x 5 2xx2 4x 5 3x2 4x 5 2x3 8x 2 10x 3x 2 12x 15
3 2 2x 11x 2x 15
Multiply 2a b3a 5b
2a b3a 5b 2a3a 5b b3a 5b
6a 2 10ab 3ab 5b2
2 2 6a 7ab 5b
Multiply x 2 2
2 x 2 x 2x 2 xx 2 2x 2
x 2 2x 2x 4
x 2 4x 4
Multiply.
109) 3x 12x 5 112) x2 3x2 3
110) 9 5x5x 9 113) t 32t 2 t 2
2 111) x 3x 5x 2 114) 3x 102
Geometry
Area of a rectangle = length x width
Perimeter of a rectangle = (2 x length) + (2 x width)
Area of a triangle = ½ x base x height
Perimeter of a triangle = side + side + side
Find the area and perimeter of the Find the area and perimeter of the rectangle triangle:
8 Area = ½ x 6 x 4=12 5 3 5 Perimeter = 5 + 5 + 6 4 = 16
Area = 3 x 8 = 24 6 Perimeter = 3 + 3 + 8 + 8 = 22
Find the area and perimeter of each figure.
115) 12
5
116)
10 8
12
Geometry Distance and Midpoint
The Distance Formula The distance between two points Ax , x and Bx , y 1 2 2 2 2 2 Distance between points A and B x2 x1 y2 y1
Example 1: Using the Distance Formula Find the distance between 3, 5 and 1,6
2 2 Distance 1 3 6 5 22 112
4 121 125 25 5
5 5
The Midpoint Formula The midpoint of the line segment joining points and is
x1 x2 y1 y2 M , 2 2
Example 2: Using the Midpoint Formula
Find the midpoint of the line segment joining points and .
3 1 5 6 1 M , = M 2, 2 2 2
Find the distance between the points Find the midpoint between the points
117) 2,5 and 2,1 . 119) and
118) 2,3 and 3, 2 120) and
Geometry Distance and Midpoint