NCERT class 10 math || Chapter -2 Polynomials || Ex-2.2 intro example | Relation zeroes coefficient

Key Concepts

• Zeros of a Polynomial: Values of 'x' that make the polynomial equal to zero.
• Coefficients of a Polynomial: The numerical or constant factor of a term in a polynomial.
• Quadratic Polynomial: A polynomial with the highest degree of 2 (ax² + bx + c).
• Alpha (α) and Beta (β): Greek letters used to represent the two zeros of a quadratic polynomial.
• Sum of the Zeros: The sum of the two zeros (α + β).
• Product of the Zeros: The product of the two zeros (α * β).
• Relationship between Zeros and Coefficients: Formulas connecting the zeros of a polynomial to its coefficients.
• Fundamental Theorem of Algebra: States that a polynomial of degree 'n' has exactly 'n' complex roots (zeros).
• Factor of a Polynomial: An expression that divides evenly into the polynomial, leaving no remainder.
• Verification of Relation: Process of confirming the relationship between zeros and coefficients using a given polynomial.
• Cubic Polynomial: A polynomial with the highest degree of 3 (ax³ + bx² + cx + d).
• Gamma (γ): Greek letter used to represent the third zero of a cubic polynomial.
• Radical Sign: The symbol (√) used to indicate a root.

Relationship Between Zeros and Coefficients of Polynomials

Zeros and Coefficients Explained

• Zeros: The values of 'x' for which the polynomial equals zero. For example, in the equation 2x + 3, the zero is -3/2 because substituting -3/2 for 'x' results in zero.
• Coefficients: The numbers that multiply the variables in a polynomial. In 2x + 3, '2' is the coefficient of 'x'.

Linear Polynomials

• For a linear polynomial like 2x + 3, setting the polynomial to zero (2x + 3 = 0) and solving for 'x' gives the zero of the polynomial.
• If x = -3/2 is a zero, then x + 3/2 is a factor of the polynomial.

Quadratic Polynomials

• Standard Form: ax² + bx + c
• Number of Zeros: A quadratic polynomial has two zeros, denoted as alpha (α) and beta (β).
• Fundamental Theorem of Algebra: This theorem explains why a polynomial of degree 'n' has 'n' zeros.

Formulas for Quadratic Polynomials

• Sum of the Zeros (α + β): -b/a
• Product of the Zeros (α * β): c/a
• Relationship: These formulas relate the zeros (α, β) to the coefficients (a, b, c) of the quadratic polynomial.

Example: x² + 7x + 10

1. Factorization: Factor the quadratic polynomial into (x + 2)(x + 5).
2. Finding Zeros: Set each factor to zero: x + 2 = 0 => x = -2 and x + 5 = 0 => x = -5. Thus, α = -2 and β = -5.
3. Sum of Zeros: α + β = -2 + (-5) = -7. Using the formula, -b/a = -7/1 = -7.
4. Product of Zeros: α * β = (-2) * (-5) = 10. Using the formula, c/a = 10/1 = 10.
5. Verification: The calculated sums and products match the values obtained using the formulas, verifying the relationship.

Alternative Representation

• -b/a can also be written as -(Coefficient of x) / (Coefficient of x²).

Proof of the Relationship

Derivation of the Formulas

1. Zeros: Let α and β be the zeros of the quadratic polynomial ax² + bx + c.
2. Factors: Then (x - α) and (x - β) are factors of the polynomial.
3. Polynomial Representation: ax² + bx + c = k(x - α)(x - β), where 'k' is a constant.
4. Expansion: Expanding the right side gives k(x² - (α + β)x + αβ).
5. Comparison: Comparing coefficients of x² , x, and the constant term:
   • a = k
   • b = -k(α + β)
   • c = kαβ
6. Solving for α + β and αβ:
   • α + β = -b/k = -b/a
   • αβ = c/k = c/a

Key Steps in the Proof

• Expressing the quadratic polynomial in terms of its factors and a constant 'k'.
• Expanding the factored form and comparing coefficients with the standard form.
• Solving for the sum and product of the zeros in terms of the coefficients.

Forming a Quadratic Polynomial from Zeros

Formula

• Given zeros α and β, the quadratic polynomial is: x² - (α + β)x + αβ

Example

• If α + β = -3 and αβ = 2, the quadratic polynomial is x² - (-3)x + 2 = x² + 3x + 2.

Proof

• The formula is derived from the factored form of the quadratic polynomial: k(x - α)(x - β). Expanding this gives k[x² - (α + β)x + αβ]. If k = 1, we get x² - (α + β)x + αβ.

Cubic Polynomials (Out of Syllabus)

Standard Form

• ax³ + bx² + cx + d

Zeros

• Three zeros: α, β, and γ

Formulas

• Sum of Zeros (α + β + γ): -b/a
• Sum of Pairwise Products (αβ + βγ + γα): c/a
• Product of Zeros (αβγ): -d/a

Note

• Cubic polynomials are not part of the 10th-grade syllabus but are included for knowledge and competitive exams.

Example 3: Finding Zeros and Verifying the Relation

Polynomial

• x² - 3

Finding Zeros

1. Set to Zero: x² - 3 = 0
2. Solve for x: x² = 3 => x = ±√3
3. Zeros: α = √3, β = -√3

Verification

1. Sum of Zeros: α + β = √3 + (-√3) = 0. Using the formula, -b/a = 0/1 = 0.
2. Product of Zeros: α * β = (√3) * (-√3) = -3. Using the formula, c/a = -3/1 = -3.

Common Mistakes

• Identifying Coefficients: Correctly identify the coefficients a, b, and c. For example, in x² - 3, b = 0 because there is no 'x' term.
• Using Sum and Product Directly: If the question provides the sum and product of zeros directly, do not calculate them again by adding or multiplying the individual zeros.

Conclusion

The video provides a detailed explanation of the relationship between the zeros and coefficients of quadratic polynomials, including proofs and examples. It emphasizes the importance of understanding the underlying concepts and avoiding common mistakes. The inclusion of cubic polynomials, although out of syllabus, adds extra knowledge for competitive exams. The key takeaway is the ability to find zeros, verify relationships, and form quadratic polynomials from given zeros or their sum and product.