CAT Concept of zeroes, sum of zeroes and product of zeroes - Practice Questions & MCQ

Definition:

1. Zeroes of a Polynomial: The values of x for which the polynomial becomes zero. In simpler terms, the roots or solutions of the equation P(x) = 0.

2. Sum of Zeroes (α and β): For a quadratic polynomial , the sum of its zeroes (roots) is given by -b/a.

3. Product of Zeroes (α and β): For the same quadratic polynomial, the product of its zeroes is c/a.

Application of Previous Concepts: 

For any polynomial, its degree can indicate the maximum number of zeroes it can have. A polynomial of degree n can have up to n zeroes.

Foundation Building Questions:

Question 1:

Given a polynomial . If α and β are the zeroes of the polynomial, what is the sum of the zeroes?

Solution: 

For the given quadratic polynomial, 

a = 2, b = -3, and c = -5.

The sum of its zeroes, α + β = -b/a = 3/2.

Question 2: 

For the polynomial , if the zeroes are α and β, find the product of the zeroes.

Solution: 

For the given polynomial, 

a = 1, b = -5, and c = 6.

The product of its zeroes, αβ = c/a = 6.

Tips and Tricks:

1. Relating Coefficients and Zeroes: For quadratic polynomials, always remember:

   - Sum of zeroes = -b/a

   - Product of zeroes = c/a

2. Factoring for Zeroes: For some quadratic polynomials, factoring can quickly give the zeroes. For example,  can be factored as (x - 1)(x - 2), indicating the zeroes are 1 and 2.

3. Degree and Zeroes: The maximum number of zeroes a polynomial can have is equal to its degree. However, not every polynomial will necessarily have that many real zeroes.

4. Polynomial Division: If you know one zero of a higher-degree polynomial, use polynomial long division or synthetic division to simplify the polynomial. This can make finding other zeroes easier.

Understanding zeroes, especially their sum and product, is vital for various problems related to polynomials, and even in higher mathematics, especially calculus and complex analysis. It provides a fundamental insight into the nature and behaviour of the polynomial.