Determine the nature of the roots of the polynomial $x^3 - 6x^2 + 11x - 6$.

Q: Determine the nature of the roots of the polynomial $x^3 - 6x^2 + 11x - 6$.

The Nature of Roots of a Polynomial Introduction \nPolynomials are algebraic expressions consisting of variables and coefficients. The nature of the roots of a polynomial can be determined using various methods, including the discriminant and the Rational Root Theorem. Discriminant Method \nThe discriminant of a polynomial is given by the formula $Delta = b^2 - 4ac$, where $a$, $b$, and $c$ are the coefficients of the polynomial. If the discriminant is positive, the polynomial has two real and distinct roots. If the discriminant is zero, the polynomial has one real root. If the discriminant is negative, the polynomial has no real roots. Rational Root Theorem \nThe Rational Root Theorem states that if a rational number $p/q$ is a root of the polynomial, then $p$ must be a factor of the constant term and $q$ must be a factor of the leading coefficient. Application to the Given Polynomial \nThe given polynomial is $x^3 - 6x^2 + 11x - 6$. We can determine the nature of its roots using the discriminant method. \nFirst, we calculate the discriminant: $$Delta = (-6)^2 - 4(1)(-6) = 36 + 24 = 60$$ \nSince the discriminant is positive, the polynomial has two real and distinct roots. Conclusion \nIn conclusion, the polynomial $x^3 - 6x^2 + 11x - 6$ has two real and distinct roots.

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The Nature of Roots of a Polynomial Introduction 
Polynomials are algebraic expressions consisting of variables and coefficients. The nature of the roots of a polynomial can be determined using various methods, including the discriminant and the Rational Root Theorem. Discriminant Method 
The discriminant of a polynomial is given by the formula $Delta = b^2 - 4ac$, where $a$, $b$, and $c$ are the coefficients of the polynomial. If the discriminant is positive, the polynomial has two real and distinct roots. If the discriminant is zero, the polynomial has one real root. If the discriminant is negative, the polynomial has no real roots. Rational Root Theorem 
The Rational Root Theorem states that if a rational number $p/q$ is a root of the polynomial, then $p$ must be a factor of the constant term and $q$ must be a factor of the leading coefficient. Application to the Given Polynomial 
The given polynomial is $x^3 - 6x^2 + 11x - 6$. We can determine the nature of its roots using the discriminant method. 
First, we calculate the discriminant: $$Delta = (-6)^2 - 4(1)(-6) = 36 + 24 = 60$$ 
Since the discriminant is positive, the polynomial has two real and distinct roots. Conclusion 
In conclusion, the polynomial $x^3 - 6x^2 + 11x - 6$ has two real and distinct roots.

MP Board · Class 10 · Mathematics · PolynomialsDetermine the nature of the roots of the polynomial $x^3 - 6x^2 + 11x - 6$.

Introduction

Discriminant Method

Rational Root Theorem

Application to the Given Polynomial

Conclusion