Understanding the Relation Between Roots and Coefficients

In competitive exams like JEE, a strong grasp of the relationship between the roots of a polynomial and its coefficients is crucial. This knowledge allows us to solve problems efficiently without explicitly finding the roots themselves. We'll explore this concept for quadratic and cubic equations, which are most commonly encountered.

Quadratic Equations: Vieta's Formulas

For a general quadratic equation of the form $ax^2 + bx + c = 0$ , where $a \neq 0$ , let the roots be $\alpha$ and $\beta$ . Vieta's formulas provide a direct link between these roots and the coefficients $a$ , $b$ , and $c$ .

Sum and product of roots for a quadratic equation.

The sum of the roots $(\alpha + \beta)$ is equal to $-b/a$ , and the product of the roots $(\alpha \beta)$ is equal to $c/a$ . This is a fundamental concept for solving many algebraic problems.

Consider the quadratic equation $ax^2 + bx + c = 0$ . If $\alpha$ and $\beta$ are its roots, then we can write the equation in factored form as $a(x - \alpha)(x - \beta) = 0$ . Expanding this, we get $a(x^2 - (\alpha + \beta)x + \alpha\beta) = 0$ , which simplifies to $ax^2 - a(\alpha + \beta)x + a\alpha\beta = 0$ . Comparing this with the original equation $ax^2 + bx + c = 0$ , we can equate the coefficients of corresponding powers of $x$ :

Coefficient of $x$ : $-a(\alpha + \beta) = b \implies \alpha + \beta = -b/a$

Constant term: $a\alpha\beta = c \implies \alpha\beta = c/a$

These are Vieta's formulas for a quadratic equation.

For the quadratic equation

2x^2 - 5x + 3 = 0

, what is the sum of its roots?

The sum of the roots is $-b/a = -(-5)/2 = 5/2$ .

For the quadratic equation

x^2 + 4x + 4 = 0

, what is the product of its roots?

The product of the roots is $c/a = 4/1 = 4$ .

Cubic Equations: Extending Vieta's Formulas

The concept extends to cubic equations. For a cubic equation of the form $ax^3 + bx^2 + cx + d = 0$ , where $a \neq 0$ , let the roots be $\alpha$ , $\beta$ , and $\gamma$ .

Sum, sum of products taken two at a time, and product of roots for a cubic equation.

For a cubic equation, the sum of roots is $-b/a$ , the sum of the products of roots taken two at a time $(\alpha\beta + \beta\gamma + \gamma\alpha)$ is $c/a$ , and the product of the roots $(\alpha\beta\gamma)$ is $-d/a$ .

Given the cubic equation $ax^3 + bx^2 + cx + d = 0$ , with roots $\alpha$ , $\beta$ , and $\gamma$ . We can express the equation in factored form as $a(x - \alpha)(x - \beta)(x - \gamma) = 0$ . Expanding this yields:

$a(x^2 - (\alpha + \beta)x + \alpha\beta)(x - \gamma) = 0$ $a(x^3 - \gamma x^2 - (\alpha + \beta)x^2 + (\alpha + \beta)\gamma x + \alpha\beta x - \alpha\beta\gamma) = 0$ $a(x^3 - (\alpha + \beta + \gamma)x^2 + (\alpha\beta + \alpha\gamma + \beta\gamma)x - \alpha\beta\gamma) = 0$ $ax^3 - a(\alpha + \beta + \gamma)x^2 + a(\alpha\beta + \alpha\gamma + \beta\gamma)x - a\alpha\beta\gamma = 0$

Comparing coefficients with $ax^3 + bx^2 + cx + d = 0$ :

Sum of roots: $\alpha + \beta + \gamma = -b/a$

Sum of products of roots taken two at a time: $\alpha\beta + \beta\gamma + \gamma\alpha = c/a$

Product of roots: $\alpha\beta\gamma = -d/a$

These are Vieta's formulas for a cubic equation.

Visualizing the relationship between roots and coefficients helps solidify understanding. For a quadratic $ax^2+bx+c=0$ with roots $\alpha, \beta$ , the relationships are $\alpha+\beta = -b/a$ and $\alpha\beta = c/a$ . For a cubic $ax^3+bx^2+cx+d=0$ with roots $\alpha, \beta, \gamma$ , the relationships are $\alpha+\beta+\gamma = -b/a$ , $\alpha\beta+\beta\gamma+\gamma\alpha = c/a$ , and $\alpha\beta\gamma = -d/a$ . These formulas are derived by expanding the factored form of the polynomial and comparing coefficients.

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For the cubic equation

x^3 - 6x^2 + 11x - 6 = 0

, what is the sum of the products of its roots taken two at a time?

The sum of the products of roots taken two at a time is $c/a = 11/1 = 11$ .

These relationships are incredibly powerful. They allow you to construct polynomials given their roots or to find properties of roots without finding them explicitly, saving significant time in exam conditions.

Applications in Problem Solving

These formulas are fundamental for solving various types of problems, including:

Finding the value of expressions involving roots (e.g., $\alpha^2 + \beta^2$ ).
Forming new equations whose roots are related to the roots of a given equation.
Analyzing the nature of roots based on coefficients.
Solving problems involving symmetric functions of roots.

Example: Finding $\alpha^2 + \beta^2$

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$ , we know $\alpha + \beta = -b/a$ and $\alpha\beta = c/a$ . We can find $\alpha^2 + \beta^2$ using the identity $(\alpha + \beta)^2 = \alpha^2 + 2\alpha\beta + \beta^2$ . Rearranging, we get $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$ . Substituting the Vieta's formulas:

$\alpha^2 + \beta^2 = (-b/a)^2 - 2(c/a) = b^2/a^2 - 2c/a = (b^2 - 2ac)/a^2$ .

\alpha

and

\beta

are roots of

x^2 - 7x + 12 = 0

, what is

\alpha^2 + \beta^2

$\alpha + \beta = 7$ , $\alpha\beta = 12$ . So, $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 7^2 - 2(12) = 49 - 24 = 25$ .

Learning Resources

Vieta's Formulas - Brilliant.org(documentation)

A comprehensive explanation of Vieta's formulas for various polynomial degrees, with examples and practice problems.

Roots of Polynomials - Khan Academy(video)

A video tutorial explaining Vieta's formulas for quadratic and cubic equations, focusing on their application.

Polynomials: Roots and Coefficients - MathWorld(documentation)

A detailed mathematical treatment of polynomial roots and their relationship to coefficients, including generalizations.

JEE Mathematics: Quadratic Equations - Byju's(blog)

Covers quadratic equations for JEE, including a section on the relation between roots and coefficients with solved examples.

Understanding Vieta's Formulas - Mathematics Stack Exchange(forum)

A discussion thread where users ask and answer questions about Vieta's formulas, offering different perspectives and problem-solving approaches.

Vieta's Formulas - Tutorialspoint(tutorial)

A step-by-step guide to understanding and applying Vieta's formulas for quadratic and cubic equations.

Polynomial Roots and Vieta's Formulas - YouTube (Unacademy JEE)(video)

A video lecture specifically tailored for JEE aspirants, explaining Vieta's formulas and their applications in exam problems.

Relation Between Roots and Coefficients of a Polynomial - Toppr(blog)

Explains the relationship between roots and coefficients for polynomials of higher degrees, with examples relevant to competitive exams.

Vieta's Formulas - Wikipedia(wikipedia)

Provides a formal definition and historical context of Vieta's formulas, including their generalization to any number of variables.

Practice Problems on Vieta's Formulas - Vedantu(tutorial)

A collection of practice problems with solutions focused on Vieta's formulas, ideal for reinforcing learning.

Relation between Roots and Coefficients