Cubic Equation 1: Roots | Algebra for CAT 2026 | Ravi Prakash Rodha

Rodha

4 chapters6 takeaways9 key terms5 questions

Overview

This video introduces cubic equations, which are polynomial equations with a degree of three. It explains the relationship between the roots (solutions) of a cubic equation and its coefficients. The presenter derives these relationships by expanding the factored form of a cubic equation and comparing it to the standard form. This concept is then extended to biquadratic (quartic) equations. Finally, the video demonstrates the application of these relationships by solving a cubic equation problem where the roots are in an arithmetic progression.

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Chapters

A cubic equation is a polynomial equation of the form ax³ + bx² + cx + d = 0.
A cubic equation has three roots, typically denoted as alpha, beta, and gamma.
If 'r' is a root, then (x - r) is a factor of the polynomial.
The expanded form of (x - alpha)(x - beta)(x - gamma) reveals the structure of a cubic equation.

Understanding the definition and structure of a cubic equation is fundamental to solving problems involving them and recognizing their properties.

The expansion of (x - alpha)(x - beta)(x - gamma) results in x³ - (alpha + beta + gamma)x² + (alpha*beta + beta*gamma + gamma*alpha)x - alpha*beta*gamma.

By comparing the expanded factored form to the standard form (after dividing by 'a'), we can establish relationships between the roots and coefficients.
The sum of the roots (alpha + beta + gamma) is equal to -b/a.
The sum of the products of the roots taken two at a time (alpha*beta + beta*gamma + gamma*alpha) is equal to c/a.
The product of the roots (alpha*beta*gamma) is equal to -d/a.

These relationships, known as Vieta's formulas for cubic equations, allow us to find information about the roots without explicitly solving the equation, and vice versa.

For the equation x³ + 2x² - 5x + 6 = 0, the sum of the roots is -2/1 = -2, the sum of products of roots taken two at a time is -5/1 = -5, and the product of the roots is -6/1 = -6.

The same principle applies to higher-degree polynomials, such as biquadratic (quartic) equations (degree 4).
A biquadratic equation has four roots (alpha, beta, gamma, delta).
The relationships between roots and coefficients follow a pattern of alternating signs and increasing combinations of roots (sum of roots, sum of products taken two at a time, sum of products taken three at a time, product of all four roots).

This demonstrates a generalizable pattern for polynomial equations, showing how the structure of the equation is intrinsically linked to its solutions.

For a biquadratic equation ax⁴ + bx³ + cx² + dx + e = 0, the sum of the roots (alpha + beta + gamma + delta) is -b/a.

When roots are in an arithmetic progression (AP), they can be represented as (a - d), a, and (a + d).
The sum of these roots (a - d) + a + (a + d) simplifies to 3a.
Using Vieta's formulas, we can equate 3a to -b/a of the cubic equation to find the value of 'a'.
Once 'a' is known, other Vieta's formulas (product of roots, sum of products taken two at a time) can be used to find unknown coefficients or other roots.

This shows a practical application of Vieta's formulas, where specific information about the nature of the roots (like being in AP) simplifies the problem significantly.

For the equation x³ - 18x² + Mx - 120 = 0, if roots are in AP, let them be (a-d), a, (a+d). Their sum is 3a = 18, so a = 6. The product (a-d)*a*(a+d) = 120. Substituting a=6 gives (6-d)*6*(6+d) = 120, leading to roots 2, 6, 10. M is the sum of products taken two at a time: (2*6) + (6*10) + (10*2) = 12 + 60 + 20 = 92.

Key takeaways

1The coefficients of a polynomial equation are directly related to the sums and products of its roots.
2Vieta's formulas provide a powerful tool for analyzing polynomial equations without direct solving.
3The structure of the expanded factored form of a polynomial reveals the pattern for Vieta's formulas.
4Understanding the relationships between roots and coefficients is crucial for solving advanced algebra problems.
5Specific conditions on roots, like being in an arithmetic progression, can simplify the application of Vieta's formulas.
6The principles for cubic equations extend to higher-degree polynomials like biquadratic equations.

Key terms

Cubic EquationRootsCoefficientsFactorsVieta's FormulasArithmetic Progression (AP)Biquadratic EquationSum of RootsProduct of Roots

Test your understanding

1What is the general form of a cubic equation and what do its roots represent?
2How does the expansion of (x - alpha)(x - beta)(x - gamma) relate to the coefficients of a cubic equation?
3What are the three main relationships defined by Vieta's formulas for a cubic equation?
4How can knowing that the roots of a cubic equation are in arithmetic progression help in solving for the roots or coefficients?
5How do the relationships between roots and coefficients for a cubic equation compare to those for a biquadratic equation?