Understanding the Theory of Equations Made Easy

The theory of equations is concerned with the algebraic structure, properties, solutions, and relationships of polynomial equations in one variable, with a central focus on their roots, coefficients, and solution methods, as systematically covered in advanced syllabus chapters such as for JEE Main.

Standard Form and Order of a Polynomial Equation

A polynomial equation in a single variable $x$ is expressed as $a_0 x^n + a_1 x^{n-1} + \ldots + a_n = 0$, where $n$ is a non-negative integer and $a_0 \neq 0$.

The highest power of $x$ determines the degree of the equation; the degree decides both the solvability and the nature of the solution set.

For $n = 2$, the equation is quadratic; for $n = 3$, it is cubic, and so forth. The fundamental theorem of algebra asserts every degree $n$ equation has exactly $n$ roots in the complex number system, counted with multiplicity.

The coefficients $a_0$, $a_1$, ..., $a_n$ may be real or complex. For the JEE Main syllabus, focus is placed on real and rational coefficients, often with attention to relationships among roots and coefficients.

Relationships Among Roots and Coefficients

Consider the general polynomial $a_0 x^n + a_1 x^{n-1} + \ldots + a_n = 0$ with roots $\alpha_1, \alpha_2, \ldots, \alpha_n$. Then, by Vieta's formulae:

The sum of the roots $\sum \alpha_i = -\dfrac{a_1}{a_0}$; the sum of the products of roots taken two at a time $\sum_{i

For a quadratic $ax^2 + bx + c = 0$, the sum is $-\dfrac{b}{a}$, and the product is $\dfrac{c}{a}$. For cubic and quartic equations, similar relationships systematically extend.

Application of Vieta's formulae enables solution of many exam problems involving symmetric functions of roots and construction of new equations from known roots.

Nature of Roots and Role of the Discriminant

For quadratic equations, the discriminant $D = b^2 - 4ac$ determines the character of roots:

• Real and distinct roots if $D > 0$
• Real and equal roots if $D = 0$
• Complex conjugate roots if $D < 0$

In higher degree equations, discriminants are more elaborate but similarly indicate multiplicity and reality of roots. For JEE Main, the quadratic discriminant remains central for inference questions; knowledge of its extension to cubics (and quartics) is occasionally required.

Problems may also involve conditions for real roots, sign patterns of coefficients, or explicit manipulation of discriminants.

For strategic preparation, see Revision Notes for additional formulae integration.

Transformation and Formation of Equations From Roots

Given original roots $\alpha_1, \alpha_2, \ldots, \alpha_n$, equations with roots of the form $k_1\alpha_1+k_2, \ldots$ or their reciprocals can be constructed. This is achieved by substituting $x$ with the inverse mapping in the original polynomial and simplifying accordingly.

If the roots of the new equation are $f(\alpha)$ where $f$ is invertible, the transformed equation is found by expressing $x$ as $f(y)$ and substituting into the original.

Such transformations appear in examination contexts involving root shifts, scalings, or reciprocals, and mastery of transformation techniques is assessed directly in several past year problems.

More comprehensive transformation concepts are discussed on the Theory Of Equations main page.

Multiplicity and Symmetry in Polynomial Roots

A root $\alpha$ of multiplicity $m$ implies $(x-\alpha)^m$ is a factor of the given polynomial. Symmetry in roots (e.g., arithmetic or geometric progression) imposes additional coefficient relationships which can be exploited for equation construction or proof-based questions.

Polynomials with real coefficients and complex roots will possess complex conjugate pairs as roots. This is frequently tested in the context of real polynomials.

When roots are in progression or bear known relations, systematic parameter substitution and use of Vieta's formulae yield required parameter constraints.

For differential approaches to polynomials, refer to Derivative Examples for further applications.

Sample Problems and Computed Solutions in Theory of Equations

Example. If the roots of $x^3 + p x^2 + q x + r = 0$ are in arithmetic progression, prove that $2p^3 - 9p q + 27r = 0$.

Let roots be $a-d,\, a,\, a+d$. Their sum is $3a = -p$, giving $a = -\dfrac{p}{3}$.

Substitute $x = a$ into the cubic: \[ (a)^3 + p(a)^2 + q(a) + r = 0 \] with $a = -\dfrac{p}{3}$.

Calculate: \[ \left(-\dfrac{p}{3}\right)^3 + p\left(-\dfrac{p}{3}\right)^2 + q\left(-\dfrac{p}{3}\right) + r = 0 \] Expanding, collect terms to give $2p^3 - 9p q + 27r = 0$.

Solution completed as above; direct stepwise expansion should be presented in examination settings.

For additional exam-style practice, see the Mock Test Series.

Example. Let $\alpha$ and $\beta$ be roots of $2x^2 + 8x + k = 0$. For $k < 0$, find the maximum value of $\left(\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha}\right)$.

Here, sum of roots $S = -\dfrac{8}{2} = -4$, product $P = \dfrac{k}{2}$. The expression \[ \dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha} = \dfrac{\alpha^2 + \beta^2}{\alpha\beta} \] with $\alpha^2+\beta^2 = S^2 - 2P$ yields \[ \dfrac{(-4)^2 - 2P}{P} = \dfrac{16 - 2P}{P} = \dfrac{16}{P} - 2 \] Since $k < 0$ implies $P < 0$, and for $P \to -\infty$, the expression has minimum value $-2$. Thus, maximum is $-2$ for negative $k$.

The above solution structure should be followed for all value-finding or maximization questions in the JEE syllabus.

Example. If $x^4 + px^2 + q = 0$ has real roots, show that $p^2 \geq 4q$ and $q \geq 0$.

Let $y = x^2$ so the equation becomes $y^2 + p y + q = 0$. For $x$ real, $y \geq 0$. The roots of the quadratic in $y$ are $y_1, y_2 = \frac{-p \pm \sqrt{p^2 - 4q}}{2}$.

For real $y$, need $p^2 - 4q \geq 0$. Also, real $x$ requires $y_1 \geq 0$ or $y_2 \geq 0$. This implies $q \geq 0$ (as product $q$ of two non-negative $y$'s is always $\geq 0$), and $p^2 \geq 4q$.

Example. If all roots of $x^2 + px + q = 0$ are positive and distinct, determine the possible signs of $p$ and $q$.

For both roots positive, sum $-p > 0$ so $p < 0$; product $q > 0$. For distinct roots, discriminant must be $> 0$.

Thus, required sign pattern is $p < 0$ and $q > 0$.

Patterns of Errors and Misconceptions in Theory of Equations

A Common Error is the confusion between sum/product of roots and their numerical evaluation: always apply correct sign based on the formulae and degree of the equation.

Students frequently misapply the discriminant for non-quadratic equations; only quadratic equations admit a direct discriminantal condition for real roots in standard form.

For broader mathematical reasoning strategies, consult Mathematical Reasoning for error minimization techniques.