ELEMENTARY ALGEBRA (CONTENTS)

ELEMENTARY ALGEBRA

H. S. HALL, M.A., and S. R. KNIGHT, B.A.

REVISED AND ENLARGED

FOR THE USE OF AMERICAN SCHOOLS

F. L. SEVENOAK, A.M.

Assistant Principal of the Academic Department,

Stevens Institute of Technology

𝔑𝔢𝔴 𝔜𝔬𝔯𝔨

THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., Ltd.

1896

By THE MACMILLAN COMPANY.

________________

First edition, 1895.

Second edition, revised and enlarged, 1896.

𝔑𝔬𝔯𝔴𝔬𝔬𝔡 𝔓𝔯𝔢𝔰𝔰

J. S. Cushing & Co. — Berwick & Smith.

Norwood Mass. U.S.A.

PREFACE.

Within a comparatively short time, the algebra requirement for admission to many of our Colleges and Schools of Science has been much increased in both thoroughness of preparation and amount of subject-matter. This increase has made necessary the rearrangement and extension of elementary algebra, and it is for this reason that the present revision of Hall and Knight's Elementary Algebra has been undertaken.

The marked success of the work, and the hearty endorsement by many of our ablest educators of the treatment of the subject as therein presented, warrant the belief that the present edition, with its additional subject-matter, will be found a desirable arrangement and satisfactory treatment "of every part of the subject required for admission to any of our Colleges or Schools of Technology.

Many changes in the original chapters have been made, among which we would call attention to the following: A proof, by mathematical induction, of the binomial theorem for positive integral index has been added to Chapter XXXIX. a method of finding a factor that will rationalize any binomial surd follows the treatment of binomial quadratic surds Chapter XLII. has been re-written in part, and appears as a chapter on equations in quadratic form ; and the chapter on logarithms lias been enlarged by the addition of a fourplace table of logarithms with explanation of its use.

Chapters XXI., XXV., XXX., XXXIII., XXXVIII., XLII., XLIII., XLIV., XLV., XLVI., XLVII., XLIX., and L. treat of portions of the subject that have not appeared in former editions. A chapter on General Theory of Equations is not usually found in an Elementary Algebra, but properly finds here a place in accordance with the purpose of the present revision; and its introduction makes the work available for use in college classes. Carefully selected exercises are given with each chapter, and at the end of the work a large miscellaneous collection will be found.

The Higher Algebra of Messrs. Hall and Knight has been drawn upon, and the works of Todliunter, Chrystal, and DeMorgan consulted in preparing the new chapters.

I gratefully acknowledge my indebtedness to Prof. J. Burkitt Webb of the Stevens Institute of Technology both for contributions of subject-matter, and valuable suggestions as to methods of treatment. My thanks are also due to Prof. W. H. Bristol, of the same institution, for suggestions as to the arrangement of the chapter on General Theory of Equations.

FRANK L. SEVENOAK.

June, 1895.

PEEFACE TO SECOND EDITION.

The printing of the present edition from entirely new plates lias enabled us to correct a few typographical errors found in the first edition, and give, at the suggestion of friends, a somewhat fuller explanation of the more difficult parts of the subject. We hope that the addition of new material to Chapters III., IV., V., X., XX., XXI., XLII., XLVIII., and of several sets of Miscellaneous Examples, will render the book still more acceptable to those whose commendation of the former edition has given us much pleasure.

June, 1896.

CONTENTS.

CHAPTER

Definitions.
Substitutions

II.

Negative Quantities.
Addition of Like Terms.

III.

Simple Brackets.
Addition.
Subtraction.
Commutative Law for Addition and Subtraction
Associative Law for Addition and Subtraction
Dimension, Degree, Ascending and Descending Power
Miscellaneous Examples I.

IV.

Multiplication
Commutative Law for Multiplication.
Associative Law for Multiplication.
Index Law for Multiplication.
Distributive Law for Multiplication
Multiplication of Compound Expressions.
Rule of Signs Note on Arithmetical and Symbolical Algebra.
Distributing the Product.
Products written by Inspection.
Multiplication by Detached Coefficients.

Division Index Law for Division.
Division of Compound Expressions Important Cases in Division.
Horner's Method of Synthetic Division

VI.

Removal and Insertion of Brackets.
Miscellaneous Examples II.

VII.

Simple Equations.

VIII.

Symbolical Expression.

IX.

Problems Leading to Simple Equations.

Resolution into Factors.
Trinomial Expressions.
Difference of Two Squares.
Important Cases.
Sum or Difference of Two Cubes.
Miscellaneous Cases Converse.
Use of Factors.
The Factor Theorem.

XI.

Highest Common Factor.
Factor Theorem Employed.

XII.

Lowest Common Multiple.

XIII.

Fractions Reduction to Lowest Terms.
Multiplication and Division.
Addition and Subtraction Rules for Change of Sign.
Cyclic Order.

XIV.

Complex Fractions.
Mixed Expressions.
Miscellaneous Exercise on Fractions.

XV.

Fractional and Literal Equations.

XVI.

Problems.
Miscellaneous Examples III.

XVII.

Simultaneous Equations.
Elimination by Addition or Subtraction.
Elimination by Substitution.
Elimination by Comparison .
Equations involving Three Unknown Quantities.
Reciprocal
Literal Simultaneous Equations

XVIII.

Problems.

XIX.

Indeterminate and Impossible Problems.
Negative Results.
Meaning of 𝑎/𝟎, 𝑎/∞, 𝟎/𝟎, ∞/∞.

XX.

Involution
To square a Multinomial.
To cube a Multinomial.
Application of Binomial Theorem.

XXI.

Evolution.
Square Root of a Multinomial.
Cube Root of a Multinomial.
Some Higher Roots .
The 𝒏th Root of a Multinomial.
Square and Cube Root of Numbers.

XXII.

The Theory of Indices.
Meaning of Fractional Exponent.
Meaning of Zero Exponent.
Meaning of Negative Exponent .

XXIII.

Surds (Radicals)
Reduction of Surds.
Addition and Subtraction of Surds.
Multiplication of Surds.
Division of Surds.
Compound Surds Factor which will rationalize Any Binomial Surd.
Properties of Quadratic Surds.
Square Root of a Binomial Surd.
Square Root of a Binomial Surd by Inspection.
Equations involving Surds.

XXIV.

Imaginary Quantities.
Imaginary Unit.
Fundamental Algebraic Operations.

XXV.

Problems.
Miscellaneous Examples IV.

XXVI.

Quadratic Equations.
Pure Quadratic Equations.
Affected Quadratic Equations.
Solution by Formula.
Solution by Factoring.
Formation of Equations with Given Roots.
Values found for the Unknown Quantity which do not satisfy the Original Equation.

XXVII.

Equations in Quadratic Form.

XXVIII.

Simultaneous Equations involving Quadratics.
Homogeneous Equations of the Same Degree
Symmetrical Equations.
Miscellaneous Cases.

XXIX.

Problems.

XXX.

Theory of Quadratic Equations.
Number of Roots.
Character of the Roots.
Relation of Roots and Coefficients.
Formation of Equations with Given Roots.
Miscellaneous Theorems.
Reminder Theorem.
Symmetry.

XXXI.

Indeterminate Equations of the First Degree.

XXXII.

Inequalities.
Miscellaneous Examples V.

XXXIII.

Ratio.
Proportion.
Transformations that may be made in a Proportion Variation.

XXXIV.

Arithmetical Progression.
Geometrical Progression.
Harmonical Progression.

XXXV.

Permutations and Combinations.

XXXVI.

Probability (Chance).
Miscellaneous Examples VI.

XXXVII.

Binomial Theorem: Proof for Positive Integral Index.
The General Term.
Simplest form of the Binominal Theorem.
Proof by Mathematical Induction.
Equal Coefficients.
Greatest Coefficients.
Sum of the Coefficients.
Expansion of Multinomials.
Application when Index is Negative or Fractional.

XXXVIII.

Logarithms.
Properties of Logarithms.
Characteristic and Mantisa.
Advantages of Common Logarithms.
Logarithms transformed from Base 𝑎 to Base 𝑏.
Logarithms in Arithmetical Calculation.
Four-Place Table of Logarithms.
Use of the Table.
Cologarithms.
Exponential Equations.

XXXIX.

Interest and Annuities.

XL.

Limiting Values and Vanishing Fractions

XLI.

Convergency and Divergency of Series
Tests for Convergency
Auxiliary Series

XLII.

Undetermined Coefficients.
Functions of Finite Dimensions.
Functions of Infinite Dimensions.
Expansion of Fractions into Series.
Expansion of Surds into Series Reversion of Series.
Partial Fractions.
The General Term.

XLIII.

Continued Fractions.
Formation of Successive.
Convergents Limits to Error.
Recurring Continued Fractions.

XLIV.

Summation of Series.
Scale of Relation.
The Sum of 𝒏 Terms.
The General Term.
The Method of Differences.
Any Required Term.
The Sum of 𝒏 Terms.
Piles of Shot and Shells.
Interpolation.

XLV.

Binomial Theorem; Proof for Any Index.

XLVI.

Exponential and Logarithmic Series.

XLVII.

Determinants Minors.
Vanishing of a Determinant.
Multiplication of a Determinant.
A Determinant expressed as the Sum of Determinants.
Simplification of Determinants.
Solution of Simultaneous Equations of Degree Signs of the Terms.
Determinant of Lower Order.

XLVIII.

Theory of Equations.
Horner's Method of Synthetic Division.
Number of the Roots.
Depression of Equations.
Formation of Equations.
Relations between the Roots and the Coefficients
Fractional Roots.
Imaginary Roots.
Transformation of Equations.
Standard Form of Reciprocal Equations Descartes' Rule of Signs.
Derived Functions.
Equal Roots.
Location of the Roots.
Sturm's Theorem and Method.
Graphical Representation of Functions
Solution of Higher Numerical Equations. Newton's Method.
The Cube Roots of Unity.
Cardan's Method.
Biquadratic Equations.
Incommensurable Roots.
Horner's Method of Approximation.
Any Root of Any Number.
Miscellaneous Examples VII.

ALGEBRA

CHAPTER I.

Definitions. Substitutions.

𝟏. Algebra treats of quantities as in Arithmetic, but with greater generality; for while the quantities used in arithmetical processes are denoted by ƒigures, which have a single definite value, algebraic quantities are denoted by symbols, which may have any value we choose to assign to them.

The symbols of quantity employed are usually the letters of our own alphabet; and, though there is no restriction as to the numerical values a symbol may represent, it is understood that in the same piece of work it keeps the same value throughout. Thus, when we say "let 𝑎 equal 𝟏," we do not mean that 𝑎 must have the value 𝟏 always, but only in the particular example we are considering. Moreover, we may operate with symbols without assigning to them any particular numerical value; indeed it is with such operations that Algebra is chiefly concerned. We begin with the definitions of Algebra, premising that the symbols +, -, ×, ÷, (), = will have the same meanings as in Arithmetic. Also, for the present, it will be assumed that all the algebraic symbols employed represent integral numbers.

𝟐. An algebraic expression is a collection of symbols; it may consist of one or more terms, which are the parts separated from each other by the signs + and —. Thus, 𝟕𝑎 + 𝟓𝑏 — 𝟑𝑐 — 𝒙 + 𝟐𝒚 is an expression consisting of five terms.

Note. When no sign precedes a term the sign + is understood.

𝟑. Expressions are either simple or compound. A simple expression consists of one term, as 𝟓𝑎. A compound expression consists of two or more terms. Compound expressions may be further distinguished. Thus an expression of two terms, as 𝟑𝑎 — 𝟐𝑏, is often called a binomial, and one of three terms, as 𝟐𝑎 + 𝟑𝑏 + 𝑐, a trinomial. Simple expressions are frequently spoken of as monomials, and compound expressions as multinomials or polynomials.

𝟒. When two or more quantities are multiplied together the result is called the product. One important difference between the notation of Arithmetic and Algebra should be here remarked. In Arithmetic the product of 𝟐 and 𝟑 is written 𝟐 × 𝟑, whereas in Algebra the product of 𝑎 and b may be written in any of the forms 𝑎 × 𝑏, 𝑎 · 𝑏, or 𝑎𝑏. The form 𝑎𝑏 is the most usual. Thus, if 𝑎 = 𝟐, 𝑏 = 𝟑, the product 𝑎𝑏 = 𝑎 × 𝑏 = 𝟐 × 𝟑 = 𝟔; but in Arithmetic 𝟐𝟑 means "twenty-three," or 𝟐 × 𝟏𝟎 + 𝟑.

Note. The beginner should carefully notice the difference between term and factor.

𝟔. When one of the factors of an expression is a numerical quantity, it is called the coefficient of the remaining factors. Thus, in the expression 𝟓 𝑎𝑏, 𝟓 is the coefficient. But the word coefficient is also used in a wider sense, and it is sometimes convenient to consider any factor, or factors, of a product as the coefficient of the remaining factors. Thus, in the product 𝟔 𝑎𝑏𝑐, 𝟔 𝑎 may be appropriately called the coefficient of 𝑏𝑐. A coefficient which is not merely numerical is sometimes called a literal coefficient.

Note. When the coefficient is unity it is usually omitted, and we write simply 𝑎, instead of 𝟏 𝑎.

𝟕. A power of a quantity is the product obtained by repeating that quantity any number of times as a factor, and is expressed by writing the number of factors to the right of the quantity and above it. Thus,

𝑎 × 𝑎 is called the second power of 𝑎, and is written a²;

𝑎 × 𝑎 × 𝑎 is called the thirg power of 𝑎, and is written a³;

and so on.

The index or exponent is the number which expresses the power of any quantity. Thus 𝟐, 𝟓, 𝟕 are respectively the indices of 𝑎², 𝑎⁵, 𝑎⁷.

Note. 𝑎² is usually read "𝑎 squared"; 𝑎³ is read "𝑎 cubed"; 𝑎⁴ is read "𝑎 to the fourth"; and so on. When the index is unity it is omitted, and we write simply 𝑎, instead of 𝑎¹. Thus 𝑎, 𝟏 𝑎, 𝑎¹, and 𝟏 𝑎¹ all have the same meaning.

The expression 𝟎⁰ is read as "zero to the power of zero" and is generally considered undefined.

𝟖. The beginner must be careful to distinguish between coefficient and index.

Ex. 𝟏. What is the difference in meaning between 𝟑𝑎 and 𝑎³ ?

By 𝟑𝑎 we mean the product of the quantities 𝟑 and 𝑎.

By 𝑎³ we mean the product of the quantities 𝑎, 𝑎, 𝑎.

Thus, if 𝑎 = 𝟒,

𝟑𝑎 = 𝟑 × 𝑎 = 𝟑 × 𝟒 = 𝟏𝟐;

𝑎³ = 𝑎 × 𝑎 × 𝑎 = 𝟒 × g × 𝟒 = 𝟔𝟒.

Ex. 𝟐. If 𝑏 = 𝟓, distinguish between 𝟒𝑏² and 𝟐𝑏⁴.

Here 𝟒𝑏² = 𝟒 × 𝑏 × 𝑏 = 𝟒 × 𝟓 × 𝟓 = 𝟏𝟎𝟎;

whereas 𝟐𝑏⁴ = 𝟐 × 𝑏 × 𝑏 × 𝑏 × 𝑏 = 𝟐 × 𝟓 × 𝟓 × 𝟓 × 𝟓 = 𝟏𝟐𝟓𝟎.

Ex. 𝟑. If 𝒙 = 𝟏, find the value of 𝟓𝒙⁴.

Here 𝟓𝒙⁴ = 𝟓 × 𝒙 × 𝒙 × 𝒙 × 𝒙 = 𝟓 × 𝟏 × 𝟏 × 𝟏 × 𝟏 = 𝟓.

Note. The beginner should observe that every power of 𝟏 is 𝟏.

Ex. 𝟒. If 𝑎 = 𝟒, 𝒙 = 𝟏, find the value of 𝟓𝒙ᵃ.

𝟓𝒙ᵃ = 𝟓 × 𝒙ᵃ = 𝟓 × 𝟏⁴ = 𝟓 × 𝟏 = 𝟓.

𝟗. The Sign of Continuation, ... , is read "and so on."

𝟏𝟎. The Sign of Deduction, ∴, is read "therefore" or "hence."

And The Sign of Induction, ∵, is read "because" or "since."

𝟏𝟏. In arithmetical multiplication the order in which the factors of a product are written is immaterial. Thus 𝟑 × 𝟒 = 𝟒 × 𝟑. In like manner in Algebra 𝑎𝑏 and 𝑏𝑎 each denote the product of the two quantities represented by the letters 𝑎 and 𝑏, and have therefore the same value. Although it is immaterial in what order the factors of a product are written, it is usual to arrange them alphabetically. Fractional coefficients which are greater than unity are usually kept in the form of improper fractions. Ex. If 𝑎 = 𝟔, 𝒙 = 𝟕, 𝒛 = 𝟓, find the value of 𝟏𝟑/𝟏𝟎𝑎𝒙𝒛. Here 𝟏𝟑/𝟏𝟎𝑎𝒙𝒛 = 𝟏𝟑/𝟏𝟎 × 𝟔 × 𝟕 × 𝟓 = 𝟐𝟕𝟑.

EXAMPLES I. 𝑎.

If 𝑎 = 𝟕, 𝑏 = 𝟐, 𝑐 = 𝟏, 𝒙 = 𝟓, 𝒚 = 𝟑, find the value of

𝟏. 𝟏𝟒𝒙.

𝟐. 𝒙³.

𝟑. 𝟑𝑎𝒙.

𝟒. 𝑎³.

𝟓. 𝟓𝑏𝒚.

𝟔. 𝑏⁵.

𝟕. 𝟑𝑏².

𝟖. 𝟐𝑎𝒙.

𝟗. 𝟔𝑐⁴.

𝟏𝟎. 𝟒𝒚³.

If 𝑎 = 𝟖, 𝑏 = 𝟓, 𝑐 = 𝟒, 𝒙 = 𝟏, 𝒚 = 𝟑, find the value of

𝟏𝟏. 𝟑𝑐².

𝟏𝟐. 𝟕𝒚³.

𝟏𝟑. 𝟓𝑎𝑏.

𝟏𝟒. 𝟗𝒙𝒚.

𝟏𝟓. 𝟖𝑏³.

𝟏𝟔. 𝟑𝒙⁵.

𝟏𝟕. 𝒙⁸.

𝟏𝟖. 𝟕𝒚⁴.

𝟏𝟗. 𝑐ˣ.

𝟐𝟎. 𝑏ʸ.

𝟐𝟏. 𝒚ᶜ.

𝟐𝟐. 𝒙ᵇ.

𝟐𝟑. 𝒚ᵇ.

𝟐𝟒. 𝑎ʸ.

𝟐𝟓. 𝑏ˣ.

If 𝑎 = 𝟓, 𝑏 = 𝟏, 𝑐 = 𝟔, 𝒙 = 𝟒, find the value of

𝟐𝟔. 𝟏/𝟒 𝒙⁴.

𝟐𝟕. 𝟓/𝟏𝟐 𝑐³.

𝟐𝟖. 𝟑/𝟖 𝒙³.

𝟐𝟗. 𝟏/𝟏𝟎 𝑎𝒙.

𝟑𝟎. 𝟑𝒙.

𝟑𝟏. 𝟐ᶜ.

𝟑𝟐. 𝟖ᵇ.

𝟑𝟑. 𝟕𝒙.

𝟑𝟒. 𝟕/𝟏𝟓 𝑎𝑐𝒙.

𝟑𝟓. 𝟏/𝟖 𝑏𝑐𝒙.

𝟏𝟐. When several different quantities are multiplied together a notation similar to that of Art. 𝟕 is adopted. Thus 𝑎𝑎𝑏𝑏𝑏𝑏𝑐𝑑𝑑𝑑 is written 𝑎²𝑏⁴𝑐𝑑³. And conversely 𝟕𝑎³𝑐𝑑² has the same meaning as 𝟕 × 𝑎 × 𝑎 × 𝑎 × 𝑐 × 𝑑 × 𝑑.

Ex. 𝟏. If 𝒙 = 𝟓, 𝒚 = 𝟑, find the value of 𝟒𝒙²𝒚³.

𝟒𝒙²𝒚³ = 𝟒 × 𝟓² × 𝟑³ = 𝟒 × 𝟐𝟓 × 𝟐𝟕 = 𝟐𝟕𝟎𝟎.

Ex. 𝟐. If 𝑎 = 𝟒, 𝑏 = 𝟗, 𝒙 = 𝟔, find the value of 𝟖𝑏𝒙²/𝟐𝟕𝑎³.

𝟖𝑏𝒙²/𝟐𝟕𝑎³ = 𝟖 × 𝟗 × 𝟔²/𝟐𝟕 × 𝟒³ = 𝟖 × 𝟗 × 𝟑𝟔/𝟐𝟕 × 𝟔𝟒 = 𝟐𝟓𝟗𝟐/𝟏𝟕𝟐𝟖 = 𝟑/𝟐 = 𝟏 𝟏/𝟐.

𝟏𝟑. If one factor of a product is equal to 𝟎, the product must be equal to 𝟎, whatever values the other factors may have. A factor is usually called a zero factor.

For instance, if 𝒙 = 𝟎, then 𝑎𝑏³𝒙𝒚² contains a zero factor. Therefore 𝑎𝑏³𝒙𝒚² = when 𝒙 = 𝟎, whatever be the values of 𝑎, 𝑏, 𝒚.

Again, if 𝑐 = 𝟎, then 𝑐³ = 𝟎; therefore 𝑎𝑏²𝑐³ = 𝟎, whatever values 𝑎 and 𝑏 may have.

Note. Every power of 𝟎 is 𝟎. And 𝑎⁰ is always read as 1, provided that 𝑎 ≠ 𝟎.

EXAMPLES I. 𝑏.

If 𝑎 = 𝟕, 𝑏 = 𝟐, 𝑐 = 𝟎, 𝒙 = 𝟓, 𝒚 = 𝟑 find the value of

𝟏. 𝟒𝑎𝒙².

𝟐. 𝑎³𝑏.

𝟑. 𝟖𝑏²𝒚.

𝟒. 𝟑𝒙𝒚².

𝟓. 𝟑/𝟒𝑏²𝒙.

𝟔. 𝟓/𝟔 𝑏³𝒚².

𝟕. 𝟐/𝟓 𝒙𝒚⁴.

𝟖. 𝑎³𝑐 .

𝟗. 𝑎²𝑐𝒚.

𝟏𝟎. 𝟖𝒙³𝒚.

If 𝑎 = 𝟐, 𝑏 = 𝟑, 𝑐 = 𝟏, 𝒑 = 𝟎, 𝒕 = 𝟒, 𝒓 = 𝟔 find the value of

𝟏𝟏. 𝟑𝑎²𝒓.

𝟏𝟐. 𝟖𝑎𝑏².

𝟏𝟑. 𝟔𝑎³𝑐/𝑏².

𝟏𝟒. 𝟒𝑐𝒓²/𝟗𝑎³.

𝟏𝟓. 𝟑𝑎²𝑏𝑐.

𝟏𝟔. 𝟓/𝟔 𝑏𝑎ʳ.

𝟏𝟕. 𝟖𝑏/𝟗𝑎ʳ.

𝟏𝟖. 𝟓𝑎ᵇ𝑐ʳ.

𝟏𝟗. 𝟐𝑎²𝒑/𝟕𝒓.

𝟐𝟎. 𝟑ᵃ𝟐ᵇ.

𝟐𝟏. 𝟐𝒓𝑎⁵.

𝟐𝟐. 𝑐ᵇ𝑏ᵗ.

𝟐𝟑. 𝟓𝑎ʳ𝑏ᵗ/𝟔𝟒𝒓ᵃ.

𝟐𝟒. 𝟐𝟕𝑎ᵗ/𝟑𝟐.

𝟐𝟓. 𝟔𝟒𝒕ʳ.

𝟏𝟒. Definition. The square root of any proposed expression is that quantity whose square, or second power, is equal to the given expression. Thus the square root of 𝟖𝟏 is 𝟗, because 𝟗² = 𝟖𝟏.

The square root of 𝑎 is denoted by ²√𝑎, or more simply √𝑎.

Similarly the cube, fourth, fifth, etc., root of any expression is that quantity whose third, fourth, fifth, etc., power is equal to the given expression.

The roots are denoted by the symbols ³√, ⁴√, ⁵√, etc.

Examples:

³√𝟐𝟕; because (∴) 𝟑³ = 𝟐𝟕

⁵√𝟑𝟐; because (∴) 𝟐⁵ = 𝟑𝟐

The symbol √ is sometimes called the radical sign.

Ex. 𝟏. Find the value of 𝟓√(𝟔 𝑎³𝑏⁴𝑐), when 𝑎 = 𝟑, 𝑏 = 𝟏, 𝑐 = 𝟖.

𝟓√(𝟔 𝑎³𝑏⁴𝑐) = 𝟓 × √(𝟔 × 𝟑³ × 𝟏⁴ × 𝟖)

= 𝟓 × √(𝟔 × 𝟐𝟕 × 𝟖) = 𝟓 × √𝟏𝟐𝟗𝟔 = 𝟓 × 𝟑𝟔 = 𝟏𝟖𝟎.

Ex. 𝟐. Find the value of ³√(𝑎𝑏⁴/𝟖𝒙³), when 𝑎 = 𝟗, 𝑏 = 𝟑, 𝒙 = 𝟓.

³√(𝑎𝑏⁴/𝟖𝒙³) = ³√(𝟗 × 𝟑⁴/𝟖 × 𝟓³) = ³√(𝟗 × 𝟖𝟏/𝟖 × 𝟏𝟐𝟓) = ³√(𝟗 × 𝟗 × 𝟗/𝟏𝟎𝟎𝟎) = 𝟗/𝟏𝟎.

EXAMPLES I. 𝑐.

If 𝑎 = 𝟖, 𝑐 = 𝟎, 𝒌 = 𝟗, 𝒙 = 𝟒, 𝒚 = 𝟏, find the value of

𝟏. √(𝟐𝑎).

𝟐. √(𝒌𝒙).

𝟑. √(𝟐𝑎𝒙).

𝟒. √(𝟐𝑎𝒌²).

𝟓. ³√(𝟑𝒌).

𝟔. ³√(𝑎𝒙³).

𝟕. ³√(𝟖𝒙³𝒚³).

𝟖. ³√(𝑐𝒚⁵).

𝟗. 𝟐𝒙√(𝟐𝑎𝒚).

𝟏𝟎. 𝟓𝒚√(𝟒𝒌𝒙).

𝟏𝟏. 𝟑𝑐√(𝒌𝒙).

𝟏𝟐. 𝟐𝒙𝒚√𝟒𝒚⁵.

𝟏𝟑. √(𝟖𝒙³/𝑎𝒌).

𝟏𝟒. √(𝟐𝟓𝑎/𝟐𝒌).

𝟏𝟓. √(𝟏𝟔𝒙/𝟒𝟗𝒚³).

𝟏𝟔. √(𝑐𝑎²/𝟏𝟔𝒌).

𝟏𝟕. ³√(𝟑𝑎/𝒌²).

𝟏𝟖. ³√(𝑎𝒙²/𝟐𝟕𝒚³).

If 𝑎 = 𝟒, 𝑏 = 𝟏, 𝑐 = 𝟐, 𝑑 = 𝟗, 𝒙 = 𝟓, 𝒚 = 𝟖, find the value of

𝟏𝟗. √𝟖𝑎𝑐.

𝟐𝟎. 𝟔√(𝟒𝑏³).

𝟐𝟏. 𝟕√(𝟓𝑑𝒙).

𝟐𝟐. √(𝑐⁵𝒚).

𝟐𝟑. √(𝟏/𝟓𝒙𝒚²).

𝟐𝟒. √(𝟏/𝟖𝑎𝑐𝑑).

𝟐𝟓. 𝟏/√(𝟗𝑎³𝑐²).

𝟐𝟔. 𝟏/√(𝟓𝑐⁴𝒙).

𝟐𝟕. ³√(𝟏/𝟒𝑐𝒙³).

𝟐𝟖. 𝟏/³√(𝟖𝑎𝑏²𝑐).

𝟐𝟗. ³√(𝟑𝒙³/𝑑²𝒚).

𝟑𝟎. ³√(𝟓𝑏⁸/𝒙⁴𝒚).

𝟑𝟏. √𝑑ᶜ.

𝟑𝟐. √𝒚ᵃ.

𝟑𝟑. √𝑏ᵈ.

𝟑𝟒. √𝑑ᵇ.

𝟏𝟓. We now proceed to find the numerical value of expressions which contain more than one term. In these each term can be dealt with singly by the rules already given, and by combining the terms the numerical value of the whole expression is explained.

𝟏𝟔. We have already, in Art. 𝟖, called attention to the importance of carefully distinguishing between coefficient and index; confusion between these is such a fruitful source of error with beginners that it may not be unnecessary once more to dwell on the distinction.

Ex. 𝟏. When 𝑐 = 𝟓, find the value of 𝑐⁴ ― 𝟒𝑐 + 𝟐𝑐³ ― 𝟑𝑐².

Here 𝑐⁴ = 𝟓⁴ = 𝟓 × 𝟓 × 𝟓 × 𝟓 = 𝟔𝟐𝟓;

𝟒𝑐 = 𝟒 × 𝟓 = 𝟐𝟎;

𝟐𝑐³ = 𝟐 × 𝑐³ = 𝟐 × 𝟓 × 𝟓 × 𝟓 = 𝟐𝟓𝟎;

𝟑𝑐² = 𝟑 × 𝑐² = 𝟑 × 𝟓 × 𝟓 = 𝟕𝟓.

Hence the value of the expression = 𝟔𝟐𝟓 — 𝟐𝟎 + 𝟐𝟓𝟎 — 𝟕𝟓 = 𝟕𝟖𝟎.

Ex. 𝟐. When 𝒑 = 𝟗, 𝒓 = 𝟔, 𝒌 = 𝟒, find the value of

𝟏/𝟑 ³√(𝒑𝒓/𝒌²) + √(𝟑𝒌 + 𝒌³ + 𝟓) ― 𝟐𝒓²/𝟗𝒌.

𝟏/𝟑 ³√(𝒑𝒓/𝒌²) + √(𝟑𝒌 + 𝒌³ + 𝟓) ― 𝟐𝒓²/𝟗𝒌 = 𝟏/𝟑 ³√(𝟓𝟒/𝟏𝟔) + √(𝟏𝟐 + 𝟔𝟒 + 𝟓) ― 𝟐×𝟑𝟔/𝟗×𝟒.

= 𝟏/𝟑 ³√(𝟐𝟕/𝟖) + √(𝟖𝟏) ― 𝟐

= 𝟏/𝟑 × 𝟑/𝟐 + 𝟗 ― 𝟐 = 𝟕 𝟏/𝟐.

𝟏𝟕. By Art. 𝟏𝟑 any term which contains a zero factor is itself zero, and may be called a zero term.

Ex. 𝟏. If 𝑎 = 𝟐, 𝑏 = 𝟎, 𝒙 = 𝟑, 𝒚 = 𝟏, find the value of

𝟒𝑎³ ― 𝑎𝑏³ + 𝟐𝒙𝒚² + 𝟑𝑎𝑏𝒙.

The expression = (𝟒 × 𝟐³) ― 𝟎 + (𝟐 × 𝟑 × 𝟏) + 𝟎.

𝟑𝟐 ― 𝟎 + 𝟔 + 𝟎 = 𝟑𝟖.

Note. The two zero terms do not affect the result.

Ex. 𝟐. Find the value of 𝟑/𝟓 𝒙² ― 𝑎²𝒚 + 𝟕𝑎𝑏𝒙 ― 𝟓/𝟐 𝒚³, when

𝑎 = 𝟓, 𝑏 = 𝟎, 𝒙 = 𝟕, 𝒚 = 𝟏.

𝟑/𝟓 𝒙² ― 𝑎²𝒚 + 𝟕𝑎𝑏𝒙 ― 𝟓/𝟐 𝒚³ =

𝟑/𝟓 (𝟕)² ― (𝟓)² × (𝟏) + 𝟕 × (𝟓) × (𝟎) × (𝟕) ― 𝟓/𝟐 × 𝟏³ =

𝟏𝟒𝟕/𝟓 ― 𝟐𝟓 + 𝟎 ― 𝟓/𝟐 = 𝟏𝟒𝟕/𝟓 ― 𝟐𝟓/𝟏 ― 𝟓/𝟐.

𝐿.𝐶.𝑀. 𝑜𝑓 𝑑𝑒𝒏𝑜𝒎𝒊𝒏𝑎𝒕𝑜𝒓𝒔 (𝟓×𝟏×𝟐) = 𝟏𝟎.

∴ 𝟏𝟒𝟕×𝟐/𝟓×𝟐 ― 𝟐𝟓×𝟏𝟎/𝟏×𝟏𝟎 ― 𝟓×𝟓/𝟐×𝟓 = 𝟐𝟗𝟒/𝟏𝟎 ― 𝟐𝟓𝟎/𝟏𝟎 ― 𝟐𝟓/𝟏𝟎 = 𝟐𝟗𝟒―𝟐𝟓𝟎―𝟐𝟓/𝟏𝟎 = 𝟏𝟗/𝟏𝟎 ∨ 𝟏 𝟗/𝟏𝟎.

Note. The zero term does not affect the result.

𝟏𝟖. In working examples the student should pay attention to the following hints:

𝟏. Too much importance cannot be attached to neatness of style and arrangement. The beginner should remember that neatness is in itself conducive to accuracy.

𝟐. The sign = should never be used except to connect quantities which are equal. Beginners should be particularly careful not to employ the sign of equality in any vague and inexact sense.

𝟑. Unless the expressions are very short the signs of equality in the steps of the work should be placed one under the other.

𝟒. It should be clearly brought out how each step follows from the one before it ; for this purpose it mil sometimes be advisable to add short verbal explanations; the importance of this will be seen later.

EXAMPLES I. 𝑑.

If 𝑎 = 𝟐, 𝑏 = 𝟑, 𝑐 = 𝟏, 𝑑 = 𝟎, find the numerical value of

𝟏. 𝟔𝑎 + 𝟓𝑏 ― 𝟖𝑐 + 𝟗𝑑.

𝟐. 𝟑𝑎 ― 𝟒𝑏 + 𝟔𝑐 + 𝟓𝑑.

𝟑. 𝟔𝑎𝑏 ― 𝟑𝑐𝑑 + 𝟐𝑑𝑎 ― 𝟓𝑐𝑏 𝟐𝑑𝑏.

𝟒. 𝑎𝑏𝑐 + 𝑏𝑐𝑑 + 𝑐𝑑𝑎 + 𝑑𝑎𝑏.

𝟓. 𝟑𝑎𝑏𝑐 ― 𝟐𝑏𝑐𝑑 + 𝟐𝑐𝑑𝑎 ― 𝟒𝑑𝑎𝑏.

𝟔. 𝟐𝑏𝑐 + 𝟑𝑐𝑑 ― 𝟒𝑑𝑎 + 𝟓𝑎𝑏.

𝟕. 𝟑𝑏𝑐𝑑 + 𝟓𝑐𝑑𝑎 ― 𝟕𝑑𝑎𝑏 + 𝑎𝑏𝑐.

𝟖. 𝑎² + 𝑏² + 𝑐² + 𝑑².

𝟗. 𝟐𝑎² + 𝟑𝑏³ ― 𝟒𝑐⁴.

𝟏𝟎. 𝑎⁴ + 𝑏⁴ ― 𝑐⁴.

If 𝑎 = 𝟏, 𝑏 = 𝟐, 𝑐 = 𝟑, 𝑑 = 𝟎, find the numerical value of

𝟏𝟏. 𝑎³ + 𝑏³ + 𝑐³ + 𝑑³.

𝟏𝟐. 𝟏/𝟐 𝑏𝑐³ ― 𝑎³ ― 𝑏³ ― 𝟑/𝟒 𝑎𝑏³𝑐.

𝟏𝟑. 𝟑𝑎𝑏𝑐 ― 𝑏²𝑐 ― 𝟔𝑎³.

𝟏𝟒. 𝟐𝑎² + 𝟐𝑏² + 𝟐𝑐² + 𝟐𝑑² ― 𝟐𝑏𝑐 ― 𝟐𝑐𝑑 ― 𝟐𝑑𝑎 ― 𝟐𝑎𝑏.

𝟏𝟓. 𝑎² + 𝟐𝑏² + 𝟐𝑐² + 𝑑² + 𝟐𝑎𝑏 + 𝟐𝑏𝑐 + 𝟐/𝟕 𝑐𝑑.

𝟏𝟔. 𝟐𝑐² + 𝟐𝑎² + 𝟐𝑏² ― 𝟒𝑐𝑏 + 𝟔𝑎𝑏𝑐𝑑.

𝟏𝟕. 𝟏𝟑𝑎² + 𝟏𝟏/𝟗 𝑐⁴ + 𝟐𝟎𝑎𝑏 ― 𝟏𝟔𝑎𝑐 ― 𝟏𝟔𝑏𝑐.

𝟏𝟖. 𝟔𝑎𝑏 ― 𝟒/𝟑 𝑎𝑐² ― 𝟐𝑎 + 𝟏/𝟖 𝑏⁴ ― 𝟑𝑑 + 𝟒/𝟗 𝑐³.

𝟏𝟗. 𝟏𝟐𝟓𝑏⁴𝑐 ― 𝟗𝑑⁵ + 𝟑𝑎𝑏𝑐²𝑑.

If 𝑎 = 𝟖, 𝑏 = 𝟐, 𝑐 = 𝟏, 𝒙 = 𝟗, 𝒚 = 𝟒, find the value of

𝟐𝟎. 𝟓/𝟑 𝑎 ― 𝟏/𝟗 𝑏³ + 𝟕/𝟖 𝒚².

𝟐𝟏. 𝟓/𝟐𝟕 𝑎𝒙 ― 𝟑𝟐/𝒚² ― 𝟔𝑎/𝑐𝒙𝒚.

𝟐𝟐. 𝟑𝑎²𝑏/𝑐𝒙𝒚² ― 𝟓𝒚/𝑎.

𝟐𝟑. ³√(𝟔𝑐𝒚⁴/𝒙²) + 𝟐√(𝟑𝑎³/𝟒𝑏³).

𝟐𝟒. ³√(𝑏𝒙𝒚) ― 𝟏/𝟖 𝑏² + 𝟖𝒙²/𝑏²𝒚.

𝟐𝟓. 𝟑/𝟒 𝑎𝑐 ― √(𝑏²/𝟗𝒚) ― ³√(𝑏𝒚/𝒙²).

𝟐𝟔. 𝟓𝑏²𝒚³/𝟏𝟐𝑎²𝒙 ― ³√(𝑎𝒙⁴/𝑏²𝒚²) + √(𝑎𝑏³/𝟑𝒙).

CHAPTER II.

Negative Quantities. Addition of Like Terms.

𝟏𝟗. In his arithmetical work the student has been accustomed to deal with numerical quantities connected by the signs + and — ; and in finding the value of an expression such as 𝟏 𝟑/𝟒 + 𝟕 𝟐/𝟑 — 𝟑 𝟏/𝟖 + 𝟔 ― 𝟒 𝟏/𝟓 he understands that the quantities to which the sign-f- is prefixed are additive, and those to which the sign — is prefixed are subtractive, while the first quantity, 𝟏 𝟑/𝟒, to which no sign is prefixed, is counted among the additive terms. The same notions prevail in Algebra; thus in using the expression 𝟕𝑎 + 𝟑𝑏 — 𝟒𝑐 — 𝟐𝑑 we understand the symbols 𝟕𝑎 and 𝟑𝑏 to be additive, while 𝟒𝑐 and 𝟐𝑑 are subtractive.

𝟐𝟎. But in Arithmetic the sum of the additive terms is always greater than the sum of the subtractive terms; and if the reverse were the case, the result would have no arithmetical meaning. In Algebra, however, not only may the sum of the subtractive terms exceed that of the additive, but a subtractive term may stand alone, and yet have a meaning quite intelligible. Hence all algebraic quantities may be divided into positive quantities and negative quantities, according as they are expressed with the sign + or the sign — ; and this is quite irrespective of any actual process of addition and subtraction.

This idea may be made clearer by one or two simple illustrations.

(𝒊) Suppose a man were to gain $𝟏𝟎𝟎 and then lose $𝟕𝟎, his total gain would be $𝟑𝟎. But if he first gains $𝟕𝟎 and then loses $𝟏𝟎𝟎 the result of his trading is a loss of $𝟑𝟎.

The corresponding algebraic statements would be

$𝟏𝟎𝟎 ― $𝟕𝟎 = + $𝟑𝟎,

$𝟕𝟎 ― $𝟏𝟎𝟎 = ― $𝟑𝟎,

and the negative quantity in the second case is interpreted as a debt, that is, a sum of money opposite in character to the positive quantity, or gain, in the first case; in fact it may be said to possess a subtractive quality which would produce its effect on other transactions, or perhaps wholly counterbalance a sum gained.

(𝒊𝒊) Suppose a man starting from a given point were to walk along a straight road 𝟏𝟎𝟎 yards forwards and then 𝟕𝟎 yards backwards, his distance from the starting-point would be 30 yards. But if he first walks 𝟕𝟎 yards forwards and then 𝟏𝟎𝟎 yards backwards his distance from the startingpoint would be 𝟑𝟎 yards, but on the opposite side of it. As before we have

𝟏𝟎𝟎 yards — 𝟕𝟎 yards = + 𝟑𝟎 yards,

𝟕𝟎 yards — 𝟏𝟎𝟎 yards = — 𝟑𝟎 yards. In each of these cases the man's absolute distance from the starting-point is the same; but by taking the positive and negative signs into account, we see that — 𝟑𝟎 is a distance from the starting-point equal in magnitude but opposite in direction to the distance represented by + 𝟑𝟎. Thus the negative sign may here be taken as indicating a reversal of direction. Many other illustrations might be chosen; but it will be sufficient here to remind the student that a subtractive quantity is always opposite in character to an additive quantity of equal absolute value.

Note. Absolute value is the value taken independently of the signs + and —. It is denoted by vertical bars, e.g., |𝒙|, where the quantity inside the bars is taken without regard to its sign.

𝟐𝟏. Definition. When any number of quantities are connected by the signs + and —, the resulting expression is called their algebraic sum. Thus 𝟏𝟏𝑎 — 𝟐𝟕𝑎 + 𝟏𝟑𝑏 — 𝟓𝑏 is an algebraic sum. This expression, however, is not, as will be shown, in its simplest form.

𝟐𝟐. Addition is the process of finding in simplest form the algebraic sum of any number of quantities.

𝟐𝟑. Like terms, or similar terms, do not differ, or differ only in their numerical coefficients. Other terms are called unlike, or dissimilar. Thus 𝟑𝑎, 𝟕𝑎; 𝟓𝑎²𝑏, 𝟐𝑎²𝑏; 𝟑𝑎³𝑏², — 𝟒𝑎³𝑏² are pairs of like terms; and 𝟒𝑎, 𝟑𝑏; 𝟕𝑎², 𝟗𝑎²𝑏 are pairs of unlike terms.

ADDITION OF LIKE TERMS.

Rule 𝐼. The sum of a number of like terms is a like term.

Rule 𝐼𝐼. If all the terms are positive, add the coefficients.

Ex. Find the value of 𝟖𝑎 + 𝟓𝑎. Here we have to increase 𝟖 like things by 𝟓 like things of the same kind, and the aggregate is 𝟏𝟑 of such things;

for instance, 𝟖 lbs. + 𝟓 lbs. = 𝟏𝟑 lbs.

Hence also, 𝟖𝑎 + 𝟓𝑎 = 𝟏𝟑𝑎

Similarly, 𝟖𝑎 + 𝟓𝑎 + 𝑎 + 𝟐𝑎 + 𝟔𝑎 = 𝟐𝟐𝑎.

Rule 𝐼𝐼𝐼. If all the terms are negative, add the coefficients numerically and prefix the minus sign to the sum.

Ex. To find the sum of — 𝟑𝒙, — 𝟓𝒙, — 𝟕𝒙, — 𝒙. Here the word sum indicates the aggregate of 𝟒 subtractive quantities of like character. In other words, we have to take away successively 𝟑, 𝟓, 𝟕, 𝟏 like things, and the result is the same as taking away 𝟑 + 𝟓 + 𝟕 + 𝟏 such things in the aggregate.

Thus the sum of — 𝟑𝒙, — 𝟓𝒙, — 𝟕𝒙, — 𝒙, is — 𝟏𝟔𝒙.

Rule 𝐼𝑉. If the terms are not all of the same sign, add together separately the coefficients of cdl the positive terms and the coefficients of all the negative terms; the difference of these two results, preceded by the sign of the greater, will give the coefficient of the sum required.

Ex. 𝟏. The sum of 𝟏𝟕𝒙 and — 𝟖𝒙 is 𝟗𝒙, for the difference of 𝟏𝟕 and 𝟖 is 𝟗, and the greater is positive.

Ex. 𝟐. To find the sum of 𝟖𝑎, — 𝟗𝑎, — 𝑎, 𝟑𝑎, 𝟒𝑎, —𝟏𝟏𝑎, 𝑎.

The sum of the coefficients of the positive terms is 𝟏𝟔.

The sum of the coefficients of the negative terms is 𝟐𝟏.

The difference of these is 𝟓, and the sign of the gieater is negative ; lience the required sum is — 𝟓𝑎.

We need not, however, adhere strictly to this rule, for the terms may be added or subtracted in the order we find most The sum convenient.

This process is called collecting terms.

Ex. 𝟑. Find the sum of 𝟐/𝟑 𝑎, 𝟑 𝑎, — 𝟏/𝟔 𝑎, ― 𝟐𝑎.

The sum = 𝑎 (𝟐/𝟑 + 𝟑 — 𝟏/𝟔 ― 𝟐)

= 𝑎 (𝟐/𝟑 + 𝟑 ― 𝟐 — 𝟏/𝟔) = 𝑎 (𝟐/𝟑 — 𝟏/𝟔 + 𝟏)

= 𝑎 (𝟒/𝟔 — 𝟏/𝟔 + 𝟏)

= 𝑎 (𝟑/𝟔 + 𝟏)

= 𝑎 (𝟏/𝟐 + 𝟏)

= 𝑎 (𝟑/𝟐)

= 𝟑/𝟐 𝑎.

= 𝟑 𝟐/𝟑 𝑎 — 𝟐 𝟏/𝟔 𝑎

= 𝟏 𝟏/𝟐

= 𝟑/𝟐 𝑎.

Note. The sum of two quantities numerically equal but with opposite signs is zero. The sum of 𝟓𝑎 and — 𝟓𝑎 is 𝟎.

EXAMPLES II.

Find the sum of

𝟏. 𝟓𝑎, 𝟕𝑎, 𝟏𝟏𝑎, 𝑎, 𝟐𝟑𝑎.

𝟐. 𝟒𝒙, 𝒙, 𝟑𝒙, 𝟕𝒙, 𝟗𝒙.

𝟑. 𝟕𝑏, 𝟏𝟎𝑏, 𝟏𝟏𝑏, 𝟗𝑏, 𝟐𝑏.

𝟒. 𝟔𝑐, 𝟖𝑐, 𝟐𝑐, 𝟏𝟓𝑐, 𝟏𝟗𝑐, 𝟏𝟎𝟎𝑐, 𝑐.

𝟓. ― 𝟑𝒙, ― 𝟓𝒙, ― 𝟏𝟏𝒙, ― 𝟕𝒙.

𝟔. ― 𝟓𝑏, ― 𝟔𝑏, ― 𝟏𝟏𝑏, ― 𝟏𝟖𝑏.

𝟕. ― 𝟑𝒚, ― 𝟕𝒚, ― 𝒚, ― 𝟐𝒚, ― 𝟒𝒚.

𝟖. ― 𝑐, ― 𝟐𝑐, ― 𝟓𝟎𝑐, ― 𝟏𝟑𝑐,

𝟗. ― 𝟏𝟏𝑏, ― 𝟓𝑏, ― 𝟑𝑏, 𝑏.

𝟏𝟎. 𝟓𝒙, ― 𝒙, ― 𝟑𝒙, 𝟐𝒙, ― 𝒙.

𝟏𝟏. 𝟐𝟔𝒚, ― 𝟏𝟏𝒚, 𝒚, ― 𝟏𝟓𝒚, ― 𝟑𝒚, 𝟐𝒚.

𝟏𝟐.𝟓 𝑓, ― 𝟗 𝑓, ― 𝟑𝑓, 𝟐𝟏𝑓, ― 𝟑𝟎𝑓.

𝟏𝟑. 𝟐𝒔, ― 𝟑𝒔, 𝒔, ― 𝒔, ― 𝟓𝒔, 𝟓𝒔.

𝟏𝟒. 𝟕𝒚, ― 𝟏𝟏𝒚, 𝟏𝟔𝒚, ― 𝟑𝒚, ― 𝟐𝒚.

𝟏𝟓. 𝟓𝒙, ― 𝟕𝒙, ― 𝟐𝒙, 𝟕𝒙, 𝟐𝒙, ―𝟓𝒙.

𝟏𝟔. 𝟕𝑎𝑏, ― 𝟑𝑎𝑏, ― 𝟓𝑎𝑏, 𝟐𝑎𝑏, 𝑎𝑏.

Find the value of

𝟏𝟕. ― 𝟗𝒙² + 𝟏𝟏𝒙² + 𝟑𝒙² ― 𝟒𝒙².

𝟏𝟖. 𝟑𝑎²𝒙 + 𝟏𝟖𝑎²𝒙 + 𝑎²𝒙.

𝟏𝟗. 𝟑𝑎³ ― 𝟕𝑎³ ― 𝟖𝑎³ + 𝟐𝑎³ ― 𝟏𝟏𝑎³.

𝟐𝟎. 𝟒𝒙³ ― 𝟓𝒙³ ― 𝟖𝒙³ ― 𝟕𝒙³.

𝟐𝟏. 𝟒𝑎²𝑏² ― 𝑎²𝑏² ― 𝟕𝑎²𝑏² + 𝟓𝑎²𝑏² ― 𝑎²𝑏².

𝟐𝟐. ― 𝟗𝒙⁴ ― 𝟒𝒙⁴ ― 𝟏𝟐𝒙⁴ + 𝟏𝟑𝒙⁴ ― 𝟕𝒙⁴.

𝟐𝟑. 𝟕𝑎𝑏𝑐𝑑 ― 𝟏𝟏𝑎𝑏𝑐𝑑 ― 𝟒𝟏𝑎𝑏𝑐𝑑 + 𝟐𝑎𝑏𝑐𝑑.

𝟐𝟒. 𝟏/𝟐 𝒙 ― 𝟏/𝟑𝒙 + 𝒙 + 𝟐/𝟑𝒙.

𝟐𝟓. 𝟑/𝟐 𝑎 + 𝟑/𝟓 𝑎 ― 𝟏/𝟐 𝑎.

𝟐𝟔. ― 𝟓𝑏 + 𝟏/𝟒 𝑏 ― 𝟑/𝟐 𝑏 + 𝟐𝑏 ― 𝟏/𝟐 𝑏 + 𝟕/𝟒 𝑏.

𝟐𝟕. ― 𝟓/𝟑 𝒙² ― 𝟐𝒙² ― 𝟐/𝟑 𝒙² + 𝒙² + 𝟏/𝟐 𝒙² + 𝟏𝟏/𝟔 𝒙².

𝟐𝟖. ― 𝑎𝑏 ―𝟏/𝟐 𝑎𝑏 ― 𝟏/𝟑 𝑎𝑏 ―𝟏/𝟒 𝑎𝑏 ― 𝟏/𝟔𝑎𝑏 + 𝑎𝑏 + 𝟓/𝟏𝟐 𝑎𝑏.

𝟐𝟗. 𝟐/𝟑 𝒙 ― 𝟑/𝟒 𝒙 + 𝟓/𝟔 𝒙 ― 𝟐𝒙 + 𝟏𝟏/𝟔 𝒙 ― 𝟏/𝟑 𝒙 + 𝒙.

𝟑𝟎. ―𝟓/𝟑 𝒙² ― 𝟑/𝟒 𝒙² ― 𝟒/𝟑 𝒙² ― 𝟏/𝟒 𝒙² ― 𝒙².

By Sohail Tahir

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