Finding Quadratic Polynomial With Transformed Zeroes

In mathematics, specifically in algebra, a common problem involves finding a new quadratic polynomial whose zeroes are related to the zeroes of a given quadratic polynomial. This article delves into a step-by-step solution for such a problem, providing a comprehensive understanding of the underlying concepts and techniques. We will explore how to utilize the relationships between the zeroes and coefficients of a quadratic polynomial to determine the coefficients of the new polynomial.

Problem Statement

Let's consider the specific problem: If α (alpha) and β (beta) are the zeroes of the quadratic polynomial f(x) = 2x² – 5x + 7, find the quadratic polynomial whose zeroes are 2α + 3β and 3α + 2β.

This problem requires us to first understand the relationship between the zeroes and coefficients of a quadratic polynomial. Then, we need to apply this knowledge to find the sum and product of the new zeroes (2α + 3β and 3α + 2β). Finally, we can construct the new quadratic polynomial using these values.

Understanding Quadratic Polynomials and Their Zeroes

A quadratic polynomial is a polynomial of degree two, generally represented as ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The zeroes of a quadratic polynomial are the values of x for which the polynomial equals zero. These zeroes are also known as the roots of the quadratic equation ax² + bx + c = 0.

For a quadratic polynomial ax² + bx + c, the sum of the zeroes (α + β) and the product of the zeroes (αβ) are related to the coefficients as follows:

• Sum of zeroes (α + β) = -b/a
• Product of zeroes (αβ) = c/a

These relationships are fundamental in solving problems involving quadratic polynomials and their zeroes. They allow us to determine the sum and product of the zeroes without actually finding the zeroes themselves. This is particularly useful when dealing with complex or irrational roots.

Step-by-Step Solution

1. Identify Coefficients and Apply Zeroes Relationships

Given the quadratic polynomial f(x) = 2x² – 5x + 7, we can identify the coefficients as follows:

• a = 2
• b = -5
• c = 7

Now, we can use the relationships between zeroes and coefficients to find the sum and product of α and β:

• Sum of zeroes (α + β) = -b/a = -(-5)/2 = 5/2
• Product of zeroes (αβ) = c/a = 7/2

These values will be crucial in determining the coefficients of the new quadratic polynomial.

2. Calculate the Sum of the New Zeroes

The new zeroes are 2α + 3β and 3α + 2β. Let's find their sum:

Sum of new zeroes = (2α + 3β) + (3α + 2β)

Combine like terms:

= 2α + 3α + 3β + 2β

= 5α + 5β

Factor out the common factor 5:

= 5(α + β)

Now, substitute the value of α + β we found earlier:

= 5(5/2)

= 25/2

Therefore, the sum of the new zeroes is 25/2.

3. Calculate the Product of the New Zeroes

Next, we need to find the product of the new zeroes:

Product of new zeroes = (2α + 3β)(3α + 2β)

Expand the expression using the distributive property (FOIL method):

= (2α)(3α) + (2α)(2β) + (3β)(3α) + (3β)(2β)

= 6α² + 4αβ + 9αβ + 6β²

Combine like terms:

= 6α² + 13αβ + 6β²

Rearrange the terms to group α² and β²:

= 6α² + 6β² + 13αβ

Factor out 6 from the first two terms:

= 6(α² + β²) + 13αβ

Now, we need to express α² + β² in terms of α + β and αβ. We can use the following identity:

(α + β)² = α² + 2αβ + β²

Rearrange the identity to solve for α² + β²:

α² + β² = (α + β)² - 2αβ

Substitute this expression back into the product equation:

Product of new zeroes = 6[(α + β)² - 2αβ] + 13αβ

Now, substitute the values of α + β and αβ we found earlier:

= 6[(5/2)² - 2(7/2)] + 13(7/2)

= 6[25/4 - 7] + 91/2

= 6[25/4 - 28/4] + 91/2

= 6[-3/4] + 91/2

= -18/4 + 91/2

= -9/2 + 91/2

= 82/2

= 41

Therefore, the product of the new zeroes is 41.

4. Construct the New Quadratic Polynomial

Now that we have the sum and product of the new zeroes, we can construct the quadratic polynomial. A general form of a quadratic polynomial with zeroes 'p' and 'q' is:

k[x² - (sum of zeroes)x + (product of zeroes)], where k is a constant.

In our case, the sum of the new zeroes is 25/2, and the product is 41. Let's substitute these values:

k[x² - (25/2)x + 41]

To eliminate the fraction, we can choose k = 2:

2[x² - (25/2)x + 41]

= 2x² - 25x + 82

Therefore, the required quadratic polynomial is 2x² - 25x + 82.

Conclusion

In this article, we have successfully found the quadratic polynomial whose zeroes are 2α + 3β and 3α + 2β, given that α and β are the zeroes of the quadratic polynomial f(x) = 2x² – 5x + 7. The solution involved understanding the relationships between the zeroes and coefficients of a quadratic polynomial, calculating the sum and product of the new zeroes, and then constructing the new polynomial using these values. This problem highlights the importance of algebraic manipulation and the application of fundamental concepts in solving mathematical problems. By following a step-by-step approach, we can effectively tackle complex problems and arrive at the correct solution. This method can be applied to various similar problems, making it a valuable tool in algebra.

Key Takeaways

• Understanding the relationship between the zeroes and coefficients of a quadratic polynomial is crucial.
• The sum of zeroes (α + β) = -b/a and the product of zeroes (αβ) = c/a.
• Algebraic manipulation and simplification are essential skills in solving these types of problems.
• Constructing a new quadratic polynomial involves finding the sum and product of the new zeroes.

By mastering these concepts and techniques, you can confidently solve problems involving quadratic polynomials and their zeroes. Practice with similar examples will further solidify your understanding and enhance your problem-solving skills. Remember, the key to success in mathematics is a combination of conceptual understanding and consistent practice.