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What is the quadratic polynomial whose sum and the product of zeroes is √2, 1/3 respectively?
- a)3x2 - 3√2x +1
- b)3x2 + 3√2x +1
- c)3x2 + 3√2x -1
- d)None of the above
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
What is the quadratic polynomial whose sum and the product of zeroes i...
Understanding the Problem
To find the quadratic polynomial based on the given sum and product of its zeroes, we can use Vieta's formulas. For a quadratic polynomial in the form of ax² + bx + c, the relationships are:
- Sum of zeroes (α + β) = -b/a
- Product of zeroes (αβ) = c/a
In this case, we have:
- Sum of zeroes = √2
- Product of zeroes = 1/3
Formulating the Polynomial
1. Using Vieta's Formulas:
- From the sum of zeroes: α + β = √2
- Hence, -b/a = √2
- From the product of zeroes: αβ = 1/3
- Hence, c/a = 1/3
2. Choosing a Value for 'a':
- Let's choose a = 3 (to eliminate fractions easily).
- Then, -b/3 = √2 implies b = -3√2.
- For c, we have c/3 = 1/3, leading to c = 1.
Constructing the Polynomial
Now substituting the values of a, b, and c into the quadratic polynomial form:
- The polynomial becomes:
3x² - 3√2x + 1
Conclusion
The required quadratic polynomial is:
3x² - 3√2x + 1
Thus, the correct answer is option 'A'.
To find the quadratic polynomial based on the given sum and product of its zeroes, we can use Vieta's formulas. For a quadratic polynomial in the form of ax² + bx + c, the relationships are:
- Sum of zeroes (α + β) = -b/a
- Product of zeroes (αβ) = c/a
In this case, we have:
- Sum of zeroes = √2
- Product of zeroes = 1/3
Formulating the Polynomial
1. Using Vieta's Formulas:
- From the sum of zeroes: α + β = √2
- Hence, -b/a = √2
- From the product of zeroes: αβ = 1/3
- Hence, c/a = 1/3
2. Choosing a Value for 'a':
- Let's choose a = 3 (to eliminate fractions easily).
- Then, -b/3 = √2 implies b = -3√2.
- For c, we have c/3 = 1/3, leading to c = 1.
Constructing the Polynomial
Now substituting the values of a, b, and c into the quadratic polynomial form:
- The polynomial becomes:
3x² - 3√2x + 1
Conclusion
The required quadratic polynomial is:
3x² - 3√2x + 1
Thus, the correct answer is option 'A'.
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Community Answer
What is the quadratic polynomial whose sum and the product of zeroes i...
Explanation: Sum of zeroes = α + β =√2
Product of zeroes = αβ = 1/3
∴ If α and β are zeroes of any quadratic polynomial, then the polynomial is;
x2 – (α+β)x + αβ
= x2 – (√2)x + (1/3)
= 3x2 - 3√2x +1
Product of zeroes = αβ = 1/3
∴ If α and β are zeroes of any quadratic polynomial, then the polynomial is;
x2 – (α+β)x + αβ
= x2 – (√2)x + (1/3)
= 3x2 - 3√2x +1
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