Chapter 1

11.  Algebraic expressions

The four operations and their signs.
The function of parentheses.
Terms versus factors.
Powers and exponents.
The order of operations.
Values and evaluations.
Evaluating algebraic expressions.

12.  Signed numbers: Positive and negative

The integers.
The algebraic sign and the absolute value.
Subtracting a larger number from a smaller.
The number line.
The negative of any number.

13.  Adding and subtracting signed numbers

"Adding" a negative number.
Naming terms. The rule for adding terms.
Subtracting a negative number.

14.  Multiplying and dividing signed numbers

The Rule of Signs.

15.  Some rules of algebra

The rule of symmetry. Commutative rules. Inverses.
Two rules for equations.

16.  Reciprocals and zero

The definition of reciprocals. The definition of division. Rules for 0.

17.  Removing grouping symbols

Parentheses. Brackets. Braces.
The relationship of  ba  to  ab.

18.  Adding like terms

19.  Linear equations

The law of inverses.
A logical sequence of statements.
Simple fractional equations.

10.  Word problems

11.  Inequalities

12.  Absolute value

Absolute value equations.
Absolute value inequalities.

Chapter 2

13.  Exponents I

Powers of a number.
Rules of exponents: When to add, when to multiply.

14.  Multiplying out: The distributive rule

15.  Multiplying binomials

Quadratic trinomials.

16.  Common factor

The definition of a polynomial in x.
Factoring polynomials.
Factoring by grouping.
Equations in which the unknown is a common factor.

17.  Factoring trinomials

Quadratics in different arguments.

18.  The square of a binomial

Perfect square trinomials.
The square of a trinomial.
Completing the square.
Geometrical algebra.

19.  The difference of two squares

Summary of Multiplying/Factoring.
Factoring by grouping.
The sum and difference of any two powers:  an ± bn.

20.  Exponents II

Negative exponents. Exponent 0. Scientific notation.

Chapter 3

21.  Algebraic fractions

Rational expressions. The principle of equivalent fractions. Reducing to lowest terms.

22.  Multiplying and dividing algebraic fractions

Complex fractions.

23.  Adding algebraic fractions

The Lowest Common Multiple (LCM) of a series of terms.

24.  Equations with fractions

Clearing of fractions.

25.  Word problems that lead to equations with fractions

The whole is equal to the sum of the parts.
Same time problem: Upstream-downstream.
Total time problem. Job problem.

Chapter 4

26.  Radicals:  Rational and irrational numbers

Square roots.
Equations x² = a, and the principal square root.
Rationalizing a denominator.
Real numbers.

27.  Simplifying radicals

Fractional radicand.

28.  Multiplying and dividing radicals

Conjugate pairs.

29.  Rational exponents

Roots of numbers. The index of a radical.
Fractional exponents.
Negative exponents.

30.  Complex, or imaginary, numbers

The square root of a negative number.
The real and imaginary components.
Conjugate pairs.

Chapter 5

31.  Rectangular coördinates

Straight lines.

32.  The Pythagorean distance formula

The distance of a point from the origin.
The distance between any two points.
A proof of the Pythagorean theorem.

33.  The equation and graph of a straight line

The equation of the first degree and its graph.
Vertical and horizontal lines.

34.  The slope of a straight line

The slope intercept form of the equation of a straight line. The general form.
Parallel and perpendicular lines.
The point-slope formula. The two-point formula.

35.  Simultaneous linear equations

The method of addition. The method of substitution. Cramer's Rule: The method of determinants.
Three equations in three unknowns.

36.  Word problems that lead to simultaneous equations

Investment problems. Mixture problems.
Upstream-downstream problems.

37.  Quadratic equations

The roots of a quadratic.
Solution by factoring.
Completing the square.
The quadratic formula.
The discriminant.
The graph of a quadratic: A parabola.

Chapter 6

38.  Logarithms

Definition. The three laws of logarithms.
Common logarithms.

Chapter 7

39.  Variation

Direct variation. The constant of proportionality.
Varies as the square. Varies inversely. Varies as the inverse square.

Copyright © 2021 Lawrence Spector

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