# Higher Algebra Hall & Knight Theory of Equations (Chapter 35) Solutions

Hall and Knight Higher Algebra Solutions for Chapter 35 ‘Theory of Equations’ are created to help you to understand the topics in this chapter. These solutions are an absolute guide for JEE and NEET aspirants as they contain all the latest tips and tricks to solve the equations. In this chapter, you will learn about the equations of n^{th} degree and the relation between roots and consecutive coefficients. This chapter is mainly about finding values as symmetrical functions of the root.

Higher Algebra by Hall & Knight Theory of Equations includes 5 exercises in this chapter comprising 131 questions in total. This chapter holds great importance in Class 12 Maths as well for competitive exams like IIT JEE. By solving the exercise questions of the chapter you will learn how to solve equations when their roots are in geometric progression or arithmetic progression, how to solve quadratic equations when one of the roots is a complex number, finding the value of f(x) for given functions, and how to transform an equation into another.

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## Important Topics for Hall and Knight Higher Algebra Solutions Chapter 35: Theory of Equations

**Introduction **

- let f(x)=P
_{0}x^{n}+P_{2}x^{n-2}+……..P_{n-1}x+P_{n }= 0 , be a rational integral equation of*nth*degree whereP_{1}, P_{2},…..P_{n}is the rational integral equation of*nth*degree - Any value of x which makes f(x)vanish is the root of the above equation.
- The roots of the equation can be real or imaginary

**The relation between the roots and the coefficients in an equation **

Let the equation : x^{n}+P_{1}x^{n-1}+P_{2}x^{n-2}+………………P_{n-1}^{ }x + P_{n }= 0 and roots be a,b,c,…….k

Then,

–P_{1 }= sum of roots;

P_{2} = sum of products of the roots taken two at a time ;

–P_{3 }= sum of the products of the roots taken three at a time;

…………………………………………………………………………

(-1)^{n}P_{n}= product of the roots.

For example, let x^{3}+P_{1}x^{2}+P_{2}+P_{3}=0 and roots be a,b,c, then

a + b + c = –P_{1}

ab + bc + ca = P_{2}

abc = –P_{3}

The relation between roots not only helps us finding roots but also in finding symmetrical functions of roots.

**Roots of an equation occur in pair in the following conditions:-**

- If equation, f(x)=0 has real coefficients, have a pair of imaginary roots a+ib and a-ib
- If an equation with rational coefficients then surd roots occur in pairs,i.e. a + √b and a – √b

**Nature of an Equation**

- If all the coefficients of the equation are positive then the equation has no positive roots
- If the coefficients of even power of x are of the same sign and others have contrary sign then the equation has no negative roots.
- If the equation only has even power of x and all the coefficients are of the same sign then the equation has no real roots.
- If the equation has only odd power of x and all the coefficients have the same sign then the equation has no real roots except x=0.
- Every equation of an odd degree has at least one real root whose sign is opposite to that of the last term.
- Every equation which is of even degree and has its last term negative has at least two real roots, one positive and one negative.

**Value of ****f(****x****+h) ,****when ****f(x) ****is a rational function of x**

In this chapter, you will also learn about how f(x) changes continuously from f(a) to f(b) and few properties associated with it.

**The Equation in Term of Roots**

- If a,b,c……..k are the roots of an equation then
*f(x)=P*_{0}(x-a)(x-b)(x-c)………(x-k)

- If an equation f(x)=0 has r roots equal to a, then the equation f'(x)=0 will have r-1 roots If a,b,c,……..k are roots of the equation f(x)=0then( this will help in finding the sum of assigned power of roots of an equation)

**Transformation of Equation**

- Sometimes an equation is simplified by transforming it into another equation roots being assigned related to the desired one.
- The transformation of equations can be done in several ways which are discussed in the exercise solutions of this chapter

**Cubic Equation**

- x
^{3 }+ Px^{2 }+ Qx + R = 0 is the general type of cubic equation. - The chapter has many many methods for solving cubic equations including Cardon’s solution and Palton’s Theory of Equations.

**Biquadratic Equations**

- The solution of the biquadratic equation was first obtained by Ferrari.
- The general equation for the biquadratic equation:-

x^{4} + 2Px^{3} +qx^{2}+ 2rx + 8 = 0

The general solution of the equation higher than degree four is not obtained. The chapter also consists of the solution of some miscellaneous equations.

**Exercise Discussion of Hall and Knight Higher Algebra Solutions Chapter 35: Theory of Equations**

- Higher Algebra By H.S. Hall and S.R. Knight Theory of Equations includes 5 exercises with 131 questions in total.
- Exercises 35.a has 21 questions and is based on finding the quadratic equations when their roots are in geometric or arithmetic progressions. It also includes questions where you have to find the sum of powers of the roots like fourth power.
- Exercise 35.b has 25 questions based on imaginary roots of quadratic equations.
- Exercise 35.c has 28 questions based on equal roots and roots between a specific range of numbers.
- Exercise 35.d has 29 questions based on the transformation of equations.
- Exercise 35.e has 28 questions based on questions of the cube and higher-order equations and cubic and biquadratic equations.

## Why Use Hall and Knight Higher Algebra Solutions Chapter 35: Theory of Equations by Instasolv?

- All questions of Hall & Knight Algebra Book Theory of Equations are discussed in a stepwise manner using simple language.
- The expert team of maths faculties on our platform has prepared the solutions based on extensive research.
- The solutions are precise and on-point. You may use these solutions in Class 12 board exam preparations and also for preparation of engineering entrance exams like IIT JEE.