The Project Gutenberg eBook of A Review of Algebra
Title: A Review of Algebra
Author: Romeyn Henry Rivenburg
Release date: January 9, 2012 [eBook #38536]
Language: English
Credits: Produced by Peter Vachuska, Alex Buie, Erica
Pfister-Altschul and the Online Distributed Proofreading
Team at http://www.pgdp.net
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
Copyright, 1914,
By ROMEYN H. RIVENBURG.
Copyright, 1914, in Great Britain.
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.
CONTENTS
- 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.
(c) Six 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.
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.
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.
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 etc.
Progressions
Development of formulas.
Insertion of means
- Arithmetical.
- Geometrical.
Binomial Theorem
Review of binomial theorem laws. See Involution.
Expansion of
Finding any term by
- key number method.
- or term method.
A REVIEW OF ALGEBRA
ORDER OF OPERATIONS, EVALUATION, PARENTHESES
Order of operations:
First of all, raising to a power and extracting a root.
Next, multiplication and division.
Last of all, addition and subtraction.
Find the value of:
1.
2.
3.
Evaluate:
4. if
5. if
6. if
7. if (Yale.)
Simplify:
8.
9.
10.
SPECIAL RULES OF MULTIPLICATION AND DIVISION
Give results by inspection:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
References: The chapter on Special Rules of Multiplication and Division in any algebra.
References: Special Rules of Multiplication and Division in the Outline in the front of the book.
CASES IN FACTORING
The number of terms in an expression usually gives the clue to the possible cases under which it may come. By applying the test for each and eliminating the possible cases one by one, the right case is readily found. Hence, the number of terms in the expression and a ready and accurate knowledge of the Cases in Factoring are the real keys to success in this vitally important part of algebra.
Case I. A common monomial factor. Applies to any number of terms.
Case II. A trinomial that is a perfect square. Three terms.
Case III. The difference of two squares.
A. Two terms.
B. Four terms.
C. Six terms.
D. An incomplete square. Three terms, and 4th powers or multiples of 4.
Case IV. A trinomial of the form Three terms.
Case V. A trinomial of the form Three terms.
Case VI.
A. The sum or difference of two cubes. Two terms.
B. The sum or difference of two like powers. Two terms.
Case VII. A common polynomial factor. Any composite number of terms.
Case VIII. The Factor Theorem. Any number of terms.
FACTORING
Review the Cases in Factoring (see Outline on preceding pages) and write out the prime factors of the following:
1.
2.
3.
4.
5.
6. (Five factors.)
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26. (M. I. T.)
27. (Princeton.)
28. (M. I. T.)
Solve by factoring:
29.
30.
31.
Reference: The chapter on Factoring in any algebra.
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE
Define H. C. F. and L. C. M.
Find by factoring the H. C. F. and L. C. M.:
1.
2. (Harvard.)
3. (College Entrance Board.)
4. (Cornell.)
5. (Yale.)
6. (Harvard.)
7. (College Entrance Board.)
8.
Pick out the H. C. F. and the L. C. M. of the following:
9.
10.
Reference: The chapter on H. C. F. and L. C. M. in any algebra.
FRACTIONS
Define: fraction, terms of a fraction, reciprocal of a number.
Look up the law of signs as it applies to fractions. Except for this, fractions in algebra are treated exactly the same as they are in arithmetic.
1. Reduce to lowest terms:
1. (a)
1. (b)
1. (c) (M. I. T.)
2. Reduce to a mixed expression:
2. (a)
2. (b)
3. Reduce to an improper fraction:
3. (a)
3. (b) qt.;
3. (c)
Add:
4.
5.
6.
Multiply:
7.
8.
Divide:
9.
10. (Yale.)
11. (Sheffield.)
Simplify:
12.
Reference: The chapter on Fractions in any algebra.
COMPLEX FRACTIONS AND FRACTIONAL EQUATIONS
Define a complex fraction.
Simplify:
1.
2.
3.
4. (Harvard.)
5. If what is the value of ? (Univ. of Penn.)
6. Simplify the expression (Cornell.)
7. Simplify
8. Solve
9. Solve
10. How much water must be added to 80 pounds of a 5 per cent salt solution to obtain a 4 per cent solution? (Yale.)
Reference: See Complex Fractions, and the first part of the chapter on Fractional Equations in any algebra.
FRACTIONAL EQUATIONS
1. Solve for each letter in turn
2. Solve and check:
3. Solve and check:
4. Solve (after looking up the special short method):
5. Solve by the special short method:
6. At what time between 8 and 9 o'clock are the hands of a watch (a) opposite each other? (b) at right angles? (c) together?
Work out (a) and state the equations for (b) and (c).
7. The formula for converting a temperature of F degrees Fahrenheit into its equivalent temperature of C degrees Centigrade is Express F in terms of C, and compute F for the values and (College Entrance Exam. Board.)
8. What is the price of eggs when 2 less for 24 cents raises the price 2 cents a dozen? (Yale.)
9. Solve
Reference: The Chapter on Fractional Equations in any algebra. Note particularly the special short methods, usually given about the middle of the chapter.
SIMULTANEOUS EQUATIONS
Note. Up to this point each topic presented has reviewed to some extent the preceding topics. For example, factoring reviews the special rules of multiplication and division; H. C. F. and L. C. M. review factoring; addition and subtraction of fractions and fractional equations review H. C. F. and L. C. M., etc. From this point on, however, the interdependence is not so marked, and miscellaneous examples illustrating the work already covered will be given very frequently in order to keep the whole subject fresh in mind.
1. Solve by three methods—addition and subtraction, substitution, and comparison:
Solve and check:
2.
3.
4. One half of A's marbles exceeds one half of B's and C's together by 2; twice B's marbles falls short of A's and C's together by 16; if C had four more marbles, he would have one fourth as many as A and B together. How many has each? (College Entrance Board.)
5. The sides of a triangle are a, b, c. Calculate the radii of the three circles having the vertices as centers, each being tangent externally to the other two. (Harvard.)
6. Solve graphically; then solve algebraically and compare results. (Use coördinate or squared paper.)
Factor:
7.
8.
9.
References: The chapters on Simultaneous Equations and Graphs in any algebra.
SIMULTANEOUS EQUATIONS AND INVOLUTION
1. Solve
1. Look up the method of solving when the unknowns are in the denominator. Should you clear of fractions?
2. Solve
3. Solve graphically and algebraically
4. Solve graphically and algebraically
Review:
5. The squares of the numbers from 1 to 25.
6. The cubes of the numbers from 1 to 12.
7. The fourth powers of the numbers from 1 to 5.
8. The fifth powers of the numbers from 1 to 3.
9. The binomial theorem laws. (See Involution.)
Expand: (Indicate first, then reduce.)
10.
11.
12.
13.
14. A train lost one sixth of its passengers at the first stop, 25 at the second stop, 20% of the remainder at the third stop, three quarters of the remainder at the fourth stop; 25 remain. What was the original number? (M. I. T.)
References: The chapter on Involution in any algebra. Also the references on the preceding page.
SQUARE ROOT
Find the square root of:
1.
2.
3. Find the square root to three terms of
4. Find the square root of 337,561.
5. Find the square root of 1823.29.
6. Find to four decimal places the square root of 1.672. (Princeton.)
7. Add
8. Find the value of:
9. Simplify
10. Solve by the short method:
11. It takes of a second for a ball to go from the pitcher to the catcher, and of a second for the catcher to handle it and get off a throw to second base. It is 90 feet from first base to second, and 130 feet from the catcher's position to second. A runner stealing second has a start of 13 feet when the ball leaves the pitcher's hand, and beats the throw to the base by of a second. The next time he tries it, he gets a start of only feet, and is caught by 6 feet. What is his rate of running, and the velocity of the catcher's throw? (Cornell.)
Reference: The chapter on Square Root in any algebra.
THEORY OF EXPONENTS
Review the proofs, for positive integral exponents, of:
I.
II.
III.
IV.
V.
VI.
To find the meaning of a fractional exponent.
Assume that Law I holds for all exponents.
If so,
Hence, is one of the three equal factors (hence the cube root) of
In the same way,
Hence, is one of the five equal factors (hence the fifth root) of
In the same way, in general,
Hence, the numerator of a fractional exponent indicates the power, the denominator indicates the root.
To find the meaning of a zero exponent.
Assume that Law II holds for all exponents.
If so, But by division,
Axiom I.
To find the meaning of a negative exponent.
Assume that Law I holds for all exponents.
If so,
Hence,