Higher Algebra Hall & Knight Miscellaneous Theorems & Examples (Chapter 34) Solutions

Hall and Knight Higher Algebra Solutions Chapter 34 ‘Miscellaneous Theorems and Examples’ have been prepared to assist you in solving these advanced level exercise questions effortlessly. In this chapter of Higher Algebra by Hall and Knight, we will review the fundamental laws of Algebra, evaluate the remainder and quotient when a function f(x) is divided by x-a. The topics that we will learn in this chapter include the method of detached coefficients, Horner’s method of synthetic division, symmetrical and alternating functions, some useful formulae, and evaluation of linear factors of a3+b3+c3-3abc and an+bn+cnwhen a+b+c=0. Other topics covered in this chapter include the proof of the identities using the property of cube roots of unity, Euler’s method of elimination, Sylvester’s dialytic method, and Bezout’s method.

Hall & Knight Algebra Mathematics book with solutions for Miscellaneous Theorems and Examples include 84 questions covered in 3 exercises. The questions will help you attain complete clarity about miscellaneous theorems. You will get acquainted with the most commonly occurring maths problem in your competitive exams like IIT JEE Main and JEE Advanced and in your higher studies. The questions can be solved only with good conceptual clarity in the topics. We advise you to take guidance from Instasolv’s Hall and Knight Higher Algebra Solution book for better exam preparation.

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Important Topics Covered under Hall and Knight Higher Algebra Solutions Chapter 34: Miscellaneous Theorems & Examples


  1. In Algebra, we need not put up new definitions and ideas rather we simply approach the algebraical problems with the knowledge of abstract arithmetic.
  2. In Arithmetical algebra, we evaluate the fundamental laws of performing operations by defining the algebraic terms in an arithmetically intelligible manner.
  3. In symbolical algebra, we regard the laws proved in the arithmetic algebra as universally correct and assume that the symbols will adhere to these laws of arithmetical algebra.

Law of Commutation

  1. We can make additions and subtractions irrespective of the order of the algebraic symbols.
  2. We can make multiplication and division irrespective of the order of the algebraic symbols.

Law of Distribution

According to this law, we can make multiplications and division over additions and subtractions.

Law of Indices

Remainder and Quotient when any rational integral function of x, say f(x) is divided by x-a

  1. If we assume f(x) to denote our required rational integral function, Q as the quotient and R as the reminder, then, f(x)=Q(x-a)+R Also, we must understand that the remainder R does not include x. Now suppose, x=a then, f(a)=Q0+R, therefore, R=f(a)
  2. To find the quotient, the following steps must be considered: Assume the function to be of n dimensions denoted asin this case, the quotient is of n-1 dimensions denoted as,Now, we should multiply and equate the coefficients of like powers of x. Therefore, in the quotient, each successive coefficient is formed by multiplying by the last coefficient so formed. Similarly, if the divisor is x+a, the same method can be used with a multiplier as -a.

Method of Detached Coefficients

In this, the problem is abridged by writing the coefficients of terms and using zero for the representation of the missing powers of x

We can also perform the division operation to find the quotient and remainder using Horner’s Method of Synthetic Division.

Symmetrical Functions

A function will be termed symmetrical when the values of the function will not be affected by the interchange of any of the pairs of these.

Alternating Functions

Are called so with respect to their variables, when the sign changes but the value remain unaltered.

Miscellaneous Identities


Euler’s method of Elimination, Sylvester’s Dialytic Method of Elimination, and Bezout’s method is discussed thoroughly in this chapter. You will get to see many illustrations in the form of examples to understand the application of these methods.

Exercise Discussion of Hall and Knight Higher Algebra Solutions Chapter 34 Miscellaneous Theorems & Examples

  1. The questions in exercise 34.a are based on finding the remainder and quotient of different functions divided by x+a and x-a.
  2. The questions in exercise 34.b consist of concepts of identities. You will also be required to prove other important results based on the identities discussed in this chapter.
  3. Exercise 34.c comprises questions based on elimination using Euler’s method of elimination, Sylvester’s dialytic method, and Bezout’s method.

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