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Page numbers followed by f indicate figures.
and the number line, 15f
Absolute value functions, 202, 202f
Addition:
of algebraic expressions, 146–150
associative property of, 9
closure property of, 8
commutative property of, 9
with like denominators, 146–148
of monomials, 88–89
of natural numbers, 1
in PEMDAS, 60–63
of polynomials, 90–91
in powers, 81
signal words and phrases for, 213–215
with unlike denominators, 148–150
of whole numbers, 2
Additive identity property, 10
Additive inverse property, 10–11
Age relationships problems, 227–228
Algebraic expressions, 64–73
adding and subtracting, like denominators, 146–148
adding and subtracting, unlike denominators, 148–150
evaluating, 66–70
parentheses and, 70–72
terminology of, 64–66
Algebraic fractions. See Rational expressions
dividing, 145–146
multiplying, 143–145
reducing to lowest terms, 139–143
Approximately equal to symbol (≈), 5
Associative property of addition, 9
Associative property of multiplication, 10
Axes, 171–173
and quadrants, 174
Base of an exponent, 44
and the product rule, 74
quantity enclosed in grouping symbols acting as, 81
and the quotient rule, 75, 76–77
Binary operation, 8
Binomials, 85–86
factoring, 134–137
as factors of trinomials, 127
FOIL method for multiplication, 99–101
multiplication of, 98
Braces, as grouping symbols, 58
Brackets, as grouping symbols, 58
Cartesian coordinate plane, 171–183, 172f
distance between points on, 176–177
and graphing functions, 201
midpoint between two points on, 177
ordered pairs in, 171–174
quadrants of, 174–176
slope of a line, 178–182
Closure property of addition, 8
Closure property of multiplication, 8–9
Coefficients, 40
negative, in the GCF, 123–125
numerical, 65
Coins, word problems with, 235–237
Commutative property of addition, 9
and subtraction, 24
Commutative property of multiplication, 9
Completing the square to solve quadratic equations, 166–167
Complex fractions, 151–152
Complex numbers in roots, 163, 167
Conditional equations, 154
Consecutive even integers, 229
Consecutive integers, 229
Consecutive numbers in word problems, 229–231
Consecutive odd integers, 229
Constants, 64–66
Coordinate axes, 171
Coordinate of a point, 2
Coordinates, 171
and functions, 195–196
Counting numbers, 1
Cube roots, 36–37
symbol for, 36
Cubes of numbers, 45–46
Decimals:
and perfect squares, 33
of rational numbers, 4–5
Denominators, 30
addition with unlike, 148–150
of rational expressions, 139
Dependent variables, 197
Diagonal of unit square, 5f
Difference of two cubes, 103, 134, 135
Difference of two squares, 103, 134, 135
Differences. See Subtraction
Discriminant of the quadratic equation, 167
Distance between two points, 176–177
Distributive property, 11, 96–97
and factoring, 122–123
Dividends, 110
Division:
of algebraic fractions, 145–146
of natural numbers, 2
in PEMDAS, 60–63
in polynomial expressions, 104
of polynomials by monomials, 110–113
of polynomials by polynomials, 113–117
of real numbers, 29–30
signal words and phrases for, 219–221
symbol for, 29
of whole numbers, 2
with zero, 29, 110, 139, 155, 220
Divisors, 110
Domain of a function, 195–197
e, 6
Elements of sets (∊), 12
Elimination, solving systems of equations with, 208–210
Equalities, signal words and phrases for, 221–223
Equations, 205–212. See also Linear equations; Quadratic equations
conditional, 154
equivalent, 155
for functions, 196–197
left sides of, 154
of lines, 189–194
right sides of, 154
true, 154
Equivalent equations, 155
Errors in signs of numbers, 111, 114, 123
Exponential expression, 44, 46
Exponentiation, 44
reciprocals and, 50
natural numbers as, 45–48
negative numbers as, 50–52, 77, 104
parts of exponential form, 45f
in PEMDAS, 60–63
powers and, 77–79
quotient rule for, 75–77, 80, 111
rational numbers as, 54–57
unit fraction as, 53–54
zero as, 48–49
Factoring, 119–120
binomials, 134–137
and distributive property, 122–123
and FOIL method, 128–132
four term polynomials, 126–127
greatest common factor, 120–123
guidelines for, 137–138
and multiplication, compared, 119
perfect trinomial squares, 133–134, 137–138
quadratic equations, solving, 165–166
quadratic trinomials, 127, 137–138
by trial and error, 127–132
greatest common, 120–123
prime, 148
of trinomials, binomials as, 127
False statements in one-variable linear equations, 154
Field properties, 8
FOIL method, 99–101
and factoring, 128–132
in complex fractions, 151
Fractions, 4
complex, simplifying, 151–152
and perfect squares, 33
Functions, 195–204
common examples of, 201–203
equations for, 196–197
notation for, 197–198
as sets of ordered pairs, 195–196
rules for, 196–197
Fundamental Principle of Rational Expressions, 139
GCF. See Greatest common factor (aGCF)
Geometric shapes in word problems, 246–247
Graphs:
of an integer, 4
of linear equations, 184–188, 189
slope-y-intercept form of a line, 186–188
systems of equations, solving with, 205, 210–211
Greatest common factor (GCF), 120–123
guidelines for factoring and, 137–138
with negative coefficients, 123–125
a quantity as, 125–126
in reducing algebraic fractions, 139–143
Greatest common monomial factors, 120–121
Grouping, factoring by, 127, 132–133
Grouping symbols, 58–60
and base of an exponent, 81
and simplifying, 106
Identities in one-variable linear equations, 154
Independent variables, 197
Index, 38
Inequalities:
symbols for, 14t, 160–161, 160f
consecutive, 229
perfect squares, 33
Intersecting lines and systems of equations, 205, 210–211
Left side of an equation, 154
Like terms, 87
addition and subtraction with, 88–89
combining, 89–90
Linear equations, 154–162
one-variable, solving, 154–159
point-slope formula for, 190–193
slope-y-intercept form for, 189–190
standard form, 154
two-variable, solving, 159–160
Linear inequalities, one-variable, solving, 160–162
Lines:
equations of, 189–194
graphing, 184
Long division, 113–117
symbol of, 114
Midpoints between two points, 177
“Minus numbers,” 25
Minus symbol, 25
prior to parentheses, 71
Mixtures in word problems, 232–235
Monomials, 84–85
addition of, 88–89
dividing polynomials by, 110–113
as like terms, 87
multiplication of, 94–96
polynomials, multiplying by, 96–97
subtraction of, 88–89
Multiplication:
of algebraic fractions, 143–145
associative property of, 10
of binomials, 98
closure property of, 8–9
commutative property of, 9
and factoring, compared, 119
of monomials, 94–96
of natural numbers, 1
with negative numbers, 28, 160
in PEMDAS, 60–63
in a power of a power, 77–78
of rational expressions, 143–145
signal words and phrases for, 217–219
of whole numbers, 2
Multiplicative identity property, 10
Multiplicative inverse property, 11
and zero, 12
and rational numbers, 4
Negative coefficients, GCFs with, 123–125
Negative numbers, 2
in algebraic expressions, 67
nth roots of, 38
rational numbers as, 4
Negative slope, 185
nth power of a base, 44–45
in rational exponents, 54
nth root, 38
Number line:
absolute value, 15f
and the Cartesian coordinate plane, 171
inequality of numbers, 14–15, 160f
negative numbers, 2
real numbers, 6f
Number relationships in word problems, 244–246
Numbers, 1–13
and coefficients, 65
of rational expressions, 139
Numerical coefficient, 65
One, as multiplicative identity, 10
One-variable linear equations, solving, 154–159
One-variable linear inequalities, solving, 160–162
Operations, order of, 58–63
in algebraic expressions, 67–70
grouping symbols, 58–60
PEMDAS, 60–63
and absolute values, 17
and subtraction, 21
Ordered pairs, 171–174, 173, 195–196
and solving systems of equations, 205
Order of operations, 58–63, 107
in algebraic expressions, 67–70
grouping symbols and, 58–60
PEMDAS, 60–63
Outputs to functions, 197, 198
Parallel lines, slope of, 181
Parentheses:
algebraic expressions and, 70–72
as grouping symbol, 58–59
opposite symbol preceding, 71
in PEMDAS, 60–63
Partial products, 99
PEMDAS, 60–63
Percents, 4
word problems with, 240–243
Perfect cubes, 36
of polynomials, 103
Perfect squares, 33
of polynomials, 103
Perfect trinomial squares, factoring, 133–134, 137–138
Perpendicular lines, slope of, 181–182
pi (π), 6
Points:
Cartesian coordinate plane, 172f
distance between two, on a plane, 176–177
Point-slope formula for a line, 190–193
Polynomial expressions, 104–109
division of, 104
identifying, 104–106
simplifying, 106–109
Polynomials, 85–86. See also Monomials
addition of, 90–91
division of, 110–118
factoring, 119–138
FOIL method for multiplication, 99–101
of four terms, factoring, 126–127
greatest common factors of, 120–126
like terms of, 87
monomials, dividing by, 110–113
monomials, multiplying by, 96–97
multiplication of, 101–103, 119
and polynomials, division by, 113–117
special products of, 103
subtraction of, 92–93
Positive numbers:
nth root of, 38
and square roots, 5
Positive slope, 185
Power of a difference, 81
Power of a product rule, 78–79, 80
Power of a quotient rule, 79, 80
Power of a sum, 81
Power to a power rule, 77–78, 80
Prime factors, 148
Principal nth root, 38–39
unit fraction exponents and, 53
Principle square root, 32, 33–34, 164
Product rule:
Products. See Multiplication
Quadratic equations, 163–170
completing the square, solving by, 166–167
factoring, solving by, 165–166
quadratic formula, solving by, 167–170
Quadratic formula, 167–170
Quadratic functions, 201, 202f
Quadratic trinomials, factoring, 127, 137–138
Quantities, 217
Quotient rule:
Quotients, 110–113. See also Division
Radicals, 38
simplifying, 39–42
Radicands, 38
Range of a function, 195–196
Rate-time-distance problems, 237–238
Rational expressions, 139. See also Algebraic expressions
Fundamental Principle of, 139
multiplication of, 143–145
as exponents, 54–57
Ratios in word problems, 231–232
division of, 29–30
as the domain of a function, 196–197
multiplication of, 26–28
nth roots of, 38
and the number line, 6f
properties of, 8–12
subtraction of, 21–24
Reciprocals:
and exponentiation, 50
multiplicative inverse property and, 11
Remainders, 110–113
Repeating decimals, 4–5
Right side of an equation, 154
Roots, 163, 167. See also Square roots
cube, 36-37
Rounding decimals, 5
Signal words and phrases, 213–224
for addition, 213–215
for division, 219–221
for equality, 221–223
for multiplication, 217–219
for subtraction, 215–217
Signed numbers, 17. See also Real numbers
Signs of numbers. See also Negative numbers
Simple interest in word problems, 243–244
Simplifying:
complex fractions, 151–152
and grouping symbols, 106
polynomial expressions, 106–109
radicals, 39–42
Slope of a line, 178–182, 184, 185
and the equation, determining, 189–192
formula for, 185
negative, 185
parallel lines, 181
perpendicular lines, 181–182
zero, 185
Slope-y-intercept form of a line, 184, 185
and the equation, determining, 189–190
graphing, 186–188
and systems of equations, solving, 210–211
Solutions:
to one-variable linear equations, 154–159
to one-variable linear inequalities, 160–162
process of finding, 225–227
of quadratic equations, 163–170
to systems of equations, 205–211
to two-variable linear equations, 159–160
to word problems, 225–227
Special products, 103
Square root functions, 202, 202f
in PEMDAS, 60
radicals of, simplifying, 39–42
Standard form:
for linear equations, 154
for linear equations, graphing, 184–186
for quadratic equations, 163
Substitution, solving systems of equations with, 206–208
Subtraction:
of algebraic expressions, 146–150
with like denominators, 146–148
of monomials, 88–89
of natural numbers, 1
opposite numbers, 21
in PEMDAS, 60–63
of polynomials, 92–93
powers of, 81
in the quotient rule, 75
of real numbers, 21–24
signal words and phrases for, 215–217
symbol for, 25
with unlike denominators, 148–150
of whole numbers, 1
Sum of two cubes, 103, 134, 135
Sum of two squares, 135
Sums. See Addition
Systems of equations, 205–212
elimination, solving by, 208–210
graphing, solving by, 205, 210–211
substitution, solving by, 206–208
Terminology:
of algebraic expressions, 64–66
of functions, 197–200
Terms of algebraic expressions, 83–85, 87
addition and subtraction with, 88–89
combining, 89–90
and factors, comparing, 119–120
like terms, 87
monomials, 84–85
Trial and error factoring, 127–132
Trial and error problem solving, 225
Trinomials, 85–86
perfect squares, factoring, 133–134, 137–138
True equations, 154
Two points formula for linear equations, 192–193
Two-variable linear equations, 159–160
Unit fraction exponents, 53–54
Units, in word problems, 227
Unlike terms, 87
addition and subtraction of, 88
Variables, 64–66
one-variable linear equations, 154–159
two-variable linear equations, 159–160
Vertical lines, 178
Vertical line test, 201
Word problems, 225–248
about work, 238–240
age relationships in, 227–228
with coins, 235–237
of consecutive numbers, 229–231
geometric shapes in, 246–247
mixtures in, 232–235
number relationships in, 244–246
with percentages, 240–243
rate-time-distance, 237–238
ratios in, 231–232
simple interest, 243–244
solving, process of, 225–227
units in, 227
Work, word problems about, 238–240
y-intercept
and the equation of a line, 189–190
and slope-y-intercept form of a line, 186
Zero, 6f
as additive identity, 10–11
as an exponent, 48–49
as an integer, 3
cube root of, 36
division with, 29, 110, 139, 155, 220
and the domain of a function, 196–197
multiplicative inverse property, 12
nth roots of, 38
slope of, 185