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If 𝛼 and 𝛽 are zeroes of a polynomial 6𝑥
2
-5𝑥 +1 then form a quadratic polynomial
whose zeroes are 𝛼² and 𝛽²?
Most Upvoted Answer
If 𝛼 and 𝛽 are zeroes of a polynomial 6𝑥 2 -5𝑥 +1 then form a quad...
Understanding the Polynomial and Its Zeroes
Given the polynomial:
6x² - 5x + 1
Let α and β be the zeroes of this polynomial.
Finding the Sum and Product of Zeroes
- The sum of the zeroes (α + β) is given by the formula -b/a:
- α + β = 5/6
- The product of the zeroes (αβ) is given by the formula c/a:
- αβ = 1/6
Forming a New Polynomial with Zeroes α² and β²
To form a quadratic polynomial whose zeroes are α² and β², we need to calculate the sum and product of the new zeroes.
Calculating the New Sum and Product
- Sum of the new zeroes (α² + β²):
- α² + β² can be expressed as (α + β)² - 2αβ.
- Substituting the values:
- α² + β² = (5/6)² - 2(1/6) = 25/36 - 2/6 = 25/36 - 12/36 = 13/36.
- Product of the new zeroes (α²β²):
- α²β² = (αβ)² = (1/6)² = 1/36.
Forming the Final Polynomial
Using the sum and product of the new zeroes, the polynomial can be formed as:
- x² - (sum of zeroes)x + (product of zeroes)
Therefore, the new polynomial is:
x² - (13/36)x + (1/36).
Final Result
To eliminate the fractions, multiply through by 36:
- The resulting polynomial is:
36x² - 13x + 1.
This is the required quadratic polynomial with zeroes α² and β².
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If 𝛼 and 𝛽 are zeroes of a polynomial 6𝑥 2 -5𝑥 +1 then form a quadratic polynomial whose zeroes are 𝛼² and 𝛽²?
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