The Sum and Product of the Roots

Oct 13, 2025 by miodeya.com 71 views

Introduction

In this problem, we are given a quadratic polynomial with zeroes 'a' and 'b'. We need to find the quadratic polynomial whose zeroes are 2α and 2β. To solve this problem, we will use the concept of the sum and product of the roots of a quadratic equation.

The Sum and Product of the Roots

The sum of the roots of a quadratic equation ax² + bx + c = 0 is given by -b/a, and the product of the roots is given by c/a.

Given Polynomial

The given polynomial is 4x² - x - 4. We can write it in the form ax² + bx + c as follows:

4x² - x - 4 = 4x² + (-1)x + (-4)

Finding the Sum and Product of the Roots

The sum of the roots of the given polynomial is -(-1)/4 = 1/4, and the product of the roots is -4/4 = -1.

Finding the Quadratic Polynomial

Let the quadratic polynomial be ax² + bx + c. We know that the sum of the roots of this polynomial is 2α + 2β, and the product of the roots is (2α)(2β).

Using the Sum and Product of the Roots

We can write the sum and product of the roots of the new polynomial as follows:

2α + 2β = 1/4 (2α)(2β) = -1

Simplifying the Equations

Simplifying the equations, we get:

α + β = 1/8 αβ = -1/4

Finding the Quadratic Polynomial

Now, we can find the quadratic polynomial using the sum and product of the roots. We know that the sum of the roots is α + β, and the product of the roots is αβ.

Writing the Quadratic Polynomial

The quadratic polynomial can be written as:

x² - (α + β)x + αβ

Substituting the values of α + β and αβ, we get:

x² - (1/8)x - 1/4

Simplifying the Quadratic Polynomial

Simplifying the quadratic polynomial, we get:

x² - (1/8)x - 1/4 = 8x² - x - 2

Conclusion

Therefore, the quadratic polynomial whose zeroes are 2α and 2β is 8x² - x - 2.

Key Takeaways

The sum of the roots of a quadratic equation is given by -b/a.
The product of the roots of a quadratic equation is given by c/a.
The sum and product of the roots can be used to find the quadratic polynomial.
The quadratic polynomial can be written as x² - (α + β)x + αβ.

Frequently Asked Questions

What is the sum of the roots of a quadratic equation?
What is the product of the roots of a quadratic equation?
How can we find the quadratic polynomial using the sum and product of the roots?
What is the quadratic polynomial whose zeroes are 2α and 2β?

References

[1] "Quadratic Equations" by Khan Academy
[2] "Sum and Product of Roots" by Math Open Reference
[3] "Quadratic Polynomial" by Wolfram MathWorld

Introduction

In this article, we will provide answers to some frequently asked questions related to the problem of finding the quadratic polynomial whose zeroes are 2α and 2β.

Q&A

Q1: What is the sum of the roots of a quadratic equation?

A1: The sum of the roots of a quadratic equation is given by -b/a.

Q2: What is the product of the roots of a quadratic equation?

A2: The product of the roots of a quadratic equation is given by c/a.

Q3: How can we find the quadratic polynomial using the sum and product of the roots?

A3: We can find the quadratic polynomial using the sum and product of the roots by writing it in the form x² - (α + β)x + αβ.

Q4: What is the quadratic polynomial whose zeroes are 2α and 2β?

A4: The quadratic polynomial whose zeroes are 2α and 2β is 8x² - x - 2.

Q5: How do we simplify the quadratic polynomial?

A5: We can simplify the quadratic polynomial by substituting the values of α + β and αβ into the expression x² - (α + β)x + αβ.

Q6: What is the final form of the quadratic polynomial?

A6: The final form of the quadratic polynomial is 8x² - x - 2.

Q7: Can we use the sum and product of the roots to find the quadratic polynomial in any case?

A7: Yes, we can use the sum and product of the roots to find the quadratic polynomial in any case.

Q8: What is the significance of the sum and product of the roots in finding the quadratic polynomial?

A8: The sum and product of the roots are essential in finding the quadratic polynomial as they provide the necessary information to write the polynomial in the required form.

Q9: Can we find the quadratic polynomial using other methods?

A9: Yes, we can find the quadratic polynomial using other methods such as factoring or using the quadratic formula.

Q10: What is the importance of finding the quadratic polynomial in mathematics?

A10: Finding the quadratic polynomial is an essential concept in mathematics as it helps in solving various problems related to quadratic equations and functions.

Conclusion

In this article, we have provided answers to some frequently asked questions related to the problem of finding the quadratic polynomial whose zeroes are 2α and 2β. We hope that this article has helped in clarifying any doubts and providing a better understanding of the concept.

Key Takeaways

The sum of the roots of a quadratic equation is given by -b/a.
The product of the roots of a quadratic equation is given by c/a.
The sum and product of the roots can be used to find the quadratic polynomial.
The quadratic polynomial can be written as x² - (α + β)x + αβ.
The final form of the quadratic polynomial is 8x² - x - 2.

Frequently Asked Questions

What is the sum of the roots of a quadratic equation?
What is the product of the roots of a quadratic equation?
How can we find the quadratic polynomial using the sum and product of the roots?
What is the quadratic polynomial whose zeroes are 2α and 2β?

References

[1] "Quadratic Equations" by Khan Academy
[2] "Sum and Product of Roots" by Math Open Reference
[3] "Quadratic Polynomial" by Wolfram MathWorld