Chapter Review

Key Terms

algebraic expression — constants and variables combined using addition, subtraction, multiplication, and division

associative property of addition — the sum of three numbers may be grouped differently without affecting the result; in symbols, \(a + \left( {b + c} \right) = \left( {a + b} \right) + c\)

associative property of multiplication — the product of three numbers may be grouped differently without affecting the result; in symbols, \(a \cdot \left( {b \cdot c} \right) = \left( {a \cdot b} \right) \cdot c\)

base — in exponential notation, the expression that is being multiplied

binomial — a polynomial containing two terms

coefficient — any real number \(a_{i}\) in a polynomial in the form \(a_{n}x^{n} + ... + a_{2}x^{2} + a_{1}x + a_{0}\)

commutative property of addition — two numbers may be added in either order without affecting the result; in symbols, \(a + b = b + a\)

commutative property of multiplication — two numbers may be multiplied in any order without affecting the result; in symbols, \(a \cdot b = b \cdot a\)

constant — a quantity that does not change value

degree — the highest power of the variable that occurs in a polynomial

difference of squares — the binomial that results when a binomial is multiplied by a binomial with the same terms, but the opposite sign

distributive property — the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, \(a \cdot \left( {b + c} \right) = a \cdot b + a \cdot c\)

equation — a mathematical statement indicating that two expressions are equal

exponent — in exponential notation, the raised number or variable that indicates how many times the base is being multiplied

exponential notation — a shorthand method of writing products of the same factor

factor by grouping — a method for factoring a trinomial in the form \(ax^{2} + bx + c\) by dividing the \(x\) term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression

formula — an equation expressing a relationship between constant and variable quantities

greatest common factor — the largest polynomial that divides evenly into each polynomial

identity property of addition — there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols, \(a + 0 = a\)

identity property of multiplication — there is a unique number, called the multiplicative identity, 1, which, when multiplied by a number, results in the original number; in symbols, \(a \cdot 1 = a\)

index — the number above the radical sign indicating the \(n\)th root

integers — the set consisting of the natural numbers, their opposites, and 0: \(\left\{ {\ldots,-3,-2,-1,0,1,2,3\operatorname{,\ldots}} \right\}\)

inverse property of addition — for every real number \(a,\) there is a unique number, called the additive inverse (or opposite), denoted \(- a,\) which, when added to the original number, results in the additive identity, 0; in symbols, \(a + \left( {- a} \right) = 0\)

inverse property of multiplication — for every non-zero real number \(a,\) there is a unique number, called the multiplicative inverse (or reciprocal), denoted \(\frac{1}{a},\) which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols, \(a \cdot \frac{1}{a} = 1\)

irrational numbers — the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers

leading coefficient — the coefficient of the leading term

leading term — the term containing the highest degree

least common denominator — the smallest multiple that two denominators have in common

monomial — a polynomial containing one term

natural numbers — the set of counting numbers: \(\left\{ {1,2,3\operatorname{,\ldots}} \right\}\)

order of operations — a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to operations

perfect square trinomial — the trinomial that results when a binomial is squared

polynomial — a sum of terms each consisting of a variable raised to a nonnegative integer power

principal \(\mathbf{n}\)th root — the number with the same sign as \(a\) that when raised to the \(n\)th power equals \(a\)

principal square root — the nonnegative square root of a number \(a\) that, when multiplied by itself, equals \(a\)

radical — the symbol used to indicate a root

radical expression — an expression containing a radical symbol

radicand — the number under the radical symbol

rational expression — the quotient of two polynomial expressions

rational numbers — the set of all numbers of the form \(\frac{m}{n},\) where \(m\) and \(n\) are integers and \(n \neq 0.\) Any rational number may be written as a fraction or a terminating or repeating decimal.

real number line — a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left.

real numbers — the sets of rational numbers and irrational numbers taken together

scientific notation — a shorthand notation for writing very large or very small numbers in the form \(a \times 10^{n}\) where \(1 \leq |a| < 10\) and \(n\) is an integer

term of a polynomial — any \(a_{i}x^{i}\) of a polynomial in the form \(a_{n}x^{n} + ... + a_{2}x^{2} + a_{1}x + a_{0}\)

trinomial — a polynomial containing three terms

variable — a quantity that may change value

whole numbers — the set consisting of 0 plus the natural numbers: \(\left\{ {0,1,2,3\operatorname{,\ldots}} \right\}\)

Key Equations

[Rules of Exponents For nonzero real numbers \(a\) and \(b\) and integers \(m\) and \(n\)]{.cell-center}
Product rule	\(a^{m} \cdot a^{n} = a^{m + n}\)
Quotient rule	\(\frac{a^{m}}{a^{n}} = a^{m - n}\)
Power rule	\(\left( a^{m} \right)^{n} = a^{m \cdot n}\)
Zero exponent rule	\(a^{0} = 1\)
Negative rule	\(a^{- n} = \frac{1}{a^{n}}\)
Power of a product rule	\(\left( {a \cdot b} \right)^{n} = a^{n} \cdot b^{n}\)
Power of a quotient rule	\(\left( \frac{a}{b} \right)^{n} = \frac{a^{n}}{b^{n}}\)

perfect square trinomial	\({(x + a)}^{2} = (x + a)(x + a) = x^{2} + 2ax + a^{2}\)
difference of squares	\((a + b)(a - b) = a^{2} - b^{2}\)

difference of squares	\(a^{2} - b^{2} = (a + b)(a - b)\)
perfect square trinomial	\(a^{2} + 2ab + b^{2} = {(a + b)}^{2}\)
sum of cubes	\(a^{3} + b^{3} = (a + b)\left( a^{2} - ab + b^{2} \right)\)
difference of cubes	\(a^{3} - b^{3} = (a - b)\left( a^{2} + ab + b^{2} \right)\)

Key Concepts

1.1 Real Numbers: Algebra Essentials

Rational numbers may be written as fractions or terminating or repeating decimals. See Example 1 and Example 2.
Determine whether a number is rational or irrational by writing it as a decimal. See Example 3.
The rational numbers and irrational numbers make up the set of real numbers. See Example 4. A number can be classified as natural, whole, integer, rational, or irrational. See Example 5.
The order of operations is used to evaluate expressions. See Example 6.
The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties. See Example 7.
Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. See Example 8. They take on a numerical value when evaluated by replacing variables with constants. See Example 9, Example 10, and Example 12
Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression. See Example 11 and Example 13.

1.2 Exponents and Scientific Notation

Products of exponential expressions with the same base can be simplified by adding exponents. See Example 1.
Quotients of exponential expressions with the same base can be simplified by subtracting exponents. See Example 2.
Powers of exponential expressions with the same base can be simplified by multiplying exponents. See Example 3.
An expression with exponent zero is defined as 1. See Example 4.
An expression with a negative exponent is defined as a reciprocal. See Example 5 and Example 6.
The power of a product of factors is the same as the product of the powers of the same factors. See Example 7.
The power of a quotient of factors is the same as the quotient of the powers of the same factors. See Example 8.
The rules for exponential expressions can be combined to simplify more complicated expressions. See Example 9.
Scientific notation uses powers of 10 to simplify very large or very small numbers. See Example 10 and Example 11.
Scientific notation may be used to simplify calculations with very large or very small numbers. See Example 12 and Example 13.

1.3 Radicals and Rational Exponents

The principal square root of a number \(a\) is the nonnegative number that when multiplied by itself equals \(a.\) See Example 1.
If \(a\) and \(b\) are nonnegative, the square root of the product \(ab\) is equal to the product of the square roots of \(a\) and \(b\) See Example 2 and Example 3.
If \(a\) and \(b\) are nonnegative, the square root of the quotient \(\frac{a}{b}\) is equal to the quotient of the square roots of \(a\) and \(b\) See Example 4 and Example 5.
We can add and subtract radical expressions if they have the same radicand and the same index. See Example 6 and Example 7.
Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See Example 8 and Example 9.
The principal \(n\)th root of \(a\) is the number with the same sign as \(a\) that when raised to the \(n\)th power equals \(a.\) These roots have the same properties as square roots. See Example 10.
Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See Example 11 and Example 12.
The properties of exponents apply to rational exponents. See Example 13.

1.4 Polynomials

A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See Example 1.
We can add and subtract polynomials by combining like terms. See Example 2 and Example 3.
To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products. See Example 4.
FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See Example 5.
Perfect square trinomials and difference of squares are special products. See Example 6 and Example 7.
Follow the same rules to work with polynomials containing several variables. See Example 8.

1.5 Factoring Polynomials

The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem. See Example 1.
Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term. See Example 2.
Trinomials can be factored using a process called factoring by grouping. See Example 3.
Perfect square trinomials and the difference of squares are special products and can be factored using equations. See Example 4 and Example 5.
The sum of cubes and the difference of cubes can be factored using equations. See Example 6 and Example 7.
Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. See Example 8.

1.6 Rational Expressions

Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See Example 1.
We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See Example 2.
To divide rational expressions, multiply by the reciprocal of the second expression. See Example 3.
Adding or subtracting rational expressions requires finding a common denominator. See Example 4 and Example 5.
Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. See Example 6.