Zeros of a polynomial are the values of the variable that make the polynomial equal to zero. For example in the polynomial x² – 5x + 6 the zeros are 2 and 3 as they satisfy the equation. Finding zeros helps solve equations and understand the behavior of the polynomial graphically.
Class 10 Maths Chapter 2 Polynomials covers key concepts like zeros of polynomials relationships between coefficients and zeros and division algorithm. It emphasizes quadratic and cubic polynomials. This chapter prepares students for solving real-life problems and strengthens algebraic skills essential for CBSE Exam 2024-25 and higher mathematical studies ensuring a solid foundation in polynomial functions and their applications.
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Step 1: Understanding the Polynomial
Polynomial: x² – 5x + 6
– Zeros are α and β
– Standard form: x² – (α + β)x + (α * β) = 0
Step 2: Vieta’s Formulas
We already know from the given polynomial:
– Sum of zeros: α + β = 5
– Product of zeros: α * β = 6
Step 3: Goal Calculation
We are trying to calculate: α² + β²
Step 4: Algebraic Manipulation
(α + β)² = α² + 2αβ + β²
α² + β² = (α + β)² – 2αβ
Step 5: Substitution
– (α + β)² = 5²
= 25
– αβ = 6
– α² + β² = (α + β)² – 2(αβ)
= 25 – 12
= 11
Mathematical Insight
This approach applies Vieta’s formulas to connect the coefficients of a quadratic with the characteristics of its zeros without actually solving for the zeros themselves.
Conclusion:
The value of α² + β² is 11.
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