A REVIEW OF ALGEBRA

BY

ROMEYN HENRY RIVENBURG, A.M.

HEAD OF THE DEPARTMENT OF MATHEMATICS
THE PEDDIE INSTITUTE, HIGHTSTOWN, N.J.

AMERICAN BOOK COMPANY
NEW YORK CINCINNATI CHICAGO

A REVIEW OF ALGEBRA.

E. P. 6

PREFACE

In most high schools the course in Elementary Algebra is finished by the end of the second year. By the senior year, most students have forgotten many of the principles, and a thorough review is necessary in order to prepare college candidates for the entrance examinations and for effective work in the freshman year in college. Recognizing this need, many schools are devoting at least two periods a week for part of the senior year to a review of algebra.

For such a review the regular textbook is inadequate. From an embarrassment of riches the teacher finds it laborious to select the proper examples, while the student wastes time in searching for scattered assignments. The object of this book is to conserve the time and effort of both teacher and student, by providing a thorough and effective review that can readily be completed, if need be, in two periods a week for a half year.

Each student is expected to use his regular textbook in algebra for reference, as he would use a dictionary,—to recall a definition, a rule, or a process that he has forgotten. He should be encouraged to think his way out wherever possible, however, and to refer to the textbook only when forced to do so as a last resort.

The definitions given in the General Outline should be reviewed as occasion arises for their use. The whole Outline can be profitably employed for rapid class reviews, by covering the part of the Outline that indicates the answer, the method, the example, or the formula, as the case may be.

The whole scheme of the book is ordinarily to have a page of problems represent a day's work. This, of course, does not apply to the Outlines or the few pages of theory, which can be covered more rapidly. By this plan, making only a part of the omissions indicated in the next paragraph, the essentials of the algebra can be readily covered, if need be, in from thirty to thirty-two lessons, thus leaving time for tests, even if only eighteen weeks, of two periods each, are allotted to the course.

If a brief course is desired, the Miscellaneous Examples (pp. 31 to 35, 50 to 52), many of the problems at the end of the book, and the College Entrance Examinations may be omitted without marring the continuity or the comprehensiveness of the review.

ROMEYN H. RIVENBURG.

PAGES
Outline of Elementary and Intermediate Algebra [7-13]
Order of Operations, Evaluation, Parentheses [14]
Special Rules of Multiplication and Division [15]
Cases in Factoring [16, 17]
Factoring [18]
Highest Common Factor and Lowest Common Multiple [19]
Fractions [20]
Complex Fractions and Fractional Equations [21, 22]
Simultaneous Equations and Involution [23, 24]
Square Root [25]
Theory of Exponents [26-28]
Radicals [29, 30]
Miscellaneous Examples, Algebra to Quadratics [31-35]
Quadratic Equations [36, 37]
The Theory of Quadratic Equations [38-41]
Outline of Simultaneous Quadratics [42, 43]
Simultaneous Quadratics [44]
Ratio and Proportion [45, 46]
Arithmetical Progression [47]
Geometrical Progression [48]
The Binomial Theorem [49]
Miscellaneous Examples, Quadratics and Beyond [50-52]
Problems—Linear Equations, Simultaneous Equations,
Problems—Quadratic Equations, Simultaneous Quadratics [53-57]
College Entrance Examinations [58-80]

OUTLINE OF ELEMENTARY AND INTERMEDIATE ALGEBRA

Important Definitions

Factors; coefficient; exponent; power; base; term; algebraic sum; similar terms; degree; homogeneous expression; linear equation; root of an equation; root of an expression; identity; conditional equation; prime quantity; highest common factor (H. C. F.); lowest common multiple (L. C. M.); involution; evolution; imaginary number; real number; rational; similar radicals; binomial surd; pure quadratic equation; affected quadratic equation; equation in the quadratic form; simultaneous linear equations; simultaneous quadratic equations; discriminant; symmetrical expression; ratio; proportion; fourth proportional; third proportional; mean proportional; arithmetic progression; geometric progression;

Special Rules for Multiplication and Division

1. Square of the sum of two quantities.

2. Square of the difference of two quantities.

3. Product of the sum and difference of two quantities.

4. Product of two binomials having a common term.

5. Product of two binomials whose corresponding terms are similar.

6. Square of a polynomial.

7. Sum of two cubes.

8. Difference of two cubes.

9. Sum or difference of two like powers.

Cases in Factoring

1. Common monomial factor.

2. Trinomial that is a perfect square.

3. The difference of two squares.

(a) Two terms.

(b) Four terms.

(d) Incomplete square.

4. Trinomial of the form

5. Trinomial of the form

6. Sum or difference of

two cubes. See "Special Rules," 7 and 8.
two like powers. See "Special Rules," 9.

7. Common polynomial factor. Grouping.

8. Factor Theorem.

H. C. F. and L. C. M.

H. C. F.

L. C. M.

Fractions

Reduction to lowest terms.

Reduction of a mixed number to an improper fraction.

Reduction of an improper fraction to a mixed number.

Addition and subtraction of fractions.

Multiplication and division of fractions.

Law of signs in division, changing signs of factors, etc.

Complex fractions.

Simultaneous Equations

Solved by

addition or subtraction.
substitution.
comparison.

Graphical representation.

Involution

Law of signs.

Binomial theorem laws.

Expansion of

monomials and fractions.
binomials.
trinomials.

Evolution

Law of signs.

Evolution of monomials and fractions.

Square root of algebraic expressions.

Square root of arithmetical numbers.

Optional

Cube root of algebraic expressions.
Cube root of arithmetical numbers.

Theory of Exponents

Proofs:

Meaning of

fractional exponent.
zero exponent.
negative exponent.

Four rules

To multiply quantities having the same base, add exponents.
To divide quantities having the same base, subtract exponents.
To raise to a power, multiply exponents.
To extract a root, divide the exponent of the power by the index of the root.

Radicals

Radical in its simplest form.

Transformation of radicals

Fraction under the radical sign.
Reduction to an entire surd.
Changing to surds of different order.
Reduction to simplest form.

Addition and subtraction of radicals.

Multiplication and division of radicals

Rationalization

Monomial denominator.
Binomial denominator.
Trinomial denominator.

Square root of a binomial surd.

Radical equations. Always check results to avoid extraneous roots.

Quadratic Equations

Pure.

Affected.

Methods of solving

Completing the square.
Formula. Developed from
Factoring.

Equations in the quadratic form.

Properties of quadratics

Then
Discriminant,
and its discussion.
Nature or character of the roots.

Simultaneous Quadratics

Case I.

One equation linear.

The other quadratic.

Case II.

Both equations homogeneous and of the second degree.

Case III.

Any two of the quantities

etc., given.

Case IV.

Both equations symmetrical or symmetrical except for sign. Usually one equation of high degree, the other of the first degree.

Case V. Special Devices

I. Solve for a compound unknown, like

etc., first.

II. Divide the equations, member by member.

III. Eliminate the quadratic terms.

Ratio and Proportion

Proportionals

mean,
third,
fourth.

Theorems

Product of means equals product of extremes.
If the product of two numbers equals the product of two other numbers, either pair, etc.
Alternation.
Inversion.
Composition.
Division.
Composition and division.
In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent, etc.

Special method of proving four quantities in proportion. Let