Forming A Quadratic Polynomial With Zeroes 3 + √2 And 3 - √2

In mathematics, a quadratic polynomial is a polynomial of degree two. It has the general form ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The zeroes (or roots) of a quadratic polynomial are the values of x for which the polynomial equals zero. Finding a quadratic polynomial when its zeroes are given is a fundamental concept in algebra. This article delves into the process of forming a quadratic polynomial when its zeroes are provided, specifically focusing on the zeroes 3 + √2 and 3 - √2. Understanding this process is crucial for students and enthusiasts of mathematics, as it reinforces key concepts in polynomial algebra and equation solving.

Understanding Quadratic Polynomials and Their Zeroes

Before diving into the specific problem, let's revisit the basics of quadratic polynomials and their zeroes. A quadratic polynomial, as mentioned earlier, is an expression of the form ax² + bx + c. The zeroes of this polynomial are the solutions to the equation ax² + bx + c = 0. These zeroes can be real or complex numbers. The relationship between the zeroes and the coefficients of the quadratic polynomial is described by Vieta's formulas. For a quadratic polynomial ax² + bx + c with zeroes α and β, the sum of the zeroes (α + β) is equal to -b/ a, and the product of the zeroes (αβ) is equal to c/ a. These relationships are fundamental in constructing a quadratic polynomial from its zeroes.

Constructing a Quadratic Polynomial from Zeroes

The process of forming a quadratic polynomial from its zeroes involves using the relationships derived from Vieta's formulas. If α and β are the zeroes of a quadratic polynomial, then the polynomial can be expressed in the form:

k(x² - (α + β)x + αβ)

where k is a non-zero constant. This form is derived from the fact that if α and β are the roots, then (x - α) and (x - β) are factors of the polynomial. Multiplying these factors gives us the quadratic expression. The constant k allows for multiple quadratic polynomials with the same zeroes, as multiplying the entire polynomial by a constant does not change its roots. This method is a cornerstone of polynomial algebra and provides a direct way to construct polynomials from their roots, emphasizing the interconnectedness of roots and coefficients.

Sum and Product of Zeroes: A Detailed Explanation

To effectively construct a quadratic polynomial from its zeroes, understanding the sum and product of zeroes is paramount. Given two zeroes, α and β, their sum (α + β) and their product (αβ) provide the necessary coefficients for the quadratic polynomial. The sum represents the negative coefficient of the x term when the leading coefficient is 1, and the product represents the constant term. This relationship stems directly from the expansion of the factored form of the quadratic polynomial, (x - α)(x - β). Expanding this, we get x² - (α + β)x + αβ, which clearly shows how the sum and product of the zeroes relate to the coefficients of the quadratic polynomial. This method not only simplifies the construction process but also provides a deeper understanding of the relationship between the roots and coefficients of polynomials, a crucial concept in advanced algebra.

Forming the Quadratic Polynomial with Zeroes 3 + √2 and 3 - √2

Now, let's apply this knowledge to form the quadratic polynomial whose zeroes are 3 + √2 and 3 - √2. We will follow the steps outlined above, utilizing the sum and product of zeroes to construct the polynomial. This practical application will solidify the theoretical concepts discussed earlier, demonstrating how to use algebraic principles to solve concrete problems. By working through this example, readers can gain a clear understanding of the methodology involved and apply it to similar problems.

Step 1: Identify the Zeroes

The zeroes are given as α = 3 + √2 and β = 3 - √2. Identifying the zeroes is the first crucial step in constructing the quadratic polynomial. These values are the solutions to the equation ax² + bx + c = 0, and they form the foundation for building the polynomial. The zeroes, in this case, are irrational numbers, which means they cannot be expressed as a simple fraction. This adds a layer of complexity to the problem but also provides an opportunity to showcase the versatility of algebraic methods in handling different types of numbers. Recognizing and correctly identifying the zeroes is essential for the subsequent steps, as they directly influence the sum and product calculations that follow.

Step 2: Calculate the Sum of the Zeroes

The sum of the zeroes is α + β = (3 + √2) + (3 - √2). Calculating the sum of the zeroes is a critical step in determining the coefficients of the quadratic polynomial. In this case, adding the zeroes (3 + √2) and (3 - √2) results in a simplification due to the presence of the √2 term and its negative counterpart. This simplification leads to a rational number, which makes the subsequent steps easier to manage. The sum of the zeroes directly relates to the coefficient of the x term in the quadratic polynomial, as per Vieta's formulas. Therefore, an accurate calculation of the sum is vital for the correct construction of the polynomial. This step highlights the importance of basic algebraic operations in solving more complex problems in polynomial algebra.

(3 + √2) + (3 - √2) = 3 + √2 + 3 - √2 = 6

Step 3: Calculate the Product of the Zeroes

The product of the zeroes is αβ = (3 + √2)(3 - √2). Determining the product of the zeroes is another essential step in forming the quadratic polynomial. In this case, the product involves multiplying two binomials that are in the form of a difference of squares. This specific form allows for a straightforward calculation using the identity (a + b)(a - b) = a² - b². Applying this identity simplifies the multiplication process and leads to a rational number as the result. The product of the zeroes corresponds to the constant term in the quadratic polynomial, according to Vieta's formulas. A precise calculation here is crucial for the accurate construction of the polynomial. This step demonstrates the application of algebraic identities in simplifying calculations and solving polynomial-related problems.

(3 + √2)(3 - √2) = 3² - (√2)² = 9 - 2 = 7

Step 4: Form the Quadratic Polynomial

Now, using the sum and product, we can form the quadratic polynomial. The general form is k(x² - (α + β)x + αβ). Substituting the values we calculated:

k(x² - 6x + 7)

where k is any non-zero constant. This step synthesizes the previous calculations to construct the desired quadratic polynomial. By substituting the sum and product of the zeroes into the general form, we create an expression that represents a family of quadratic polynomials with the given zeroes. The constant k allows for an infinite number of polynomials that satisfy the condition, as multiplying a polynomial by a constant does not alter its roots. For simplicity, k is often taken as 1, resulting in the simplest form of the polynomial. This final step underscores the connection between the zeroes and the coefficients of the polynomial, demonstrating the power of algebraic methods in problem-solving.

Step 5: Simplify (Optional)

If we let k = 1, the quadratic polynomial is x² - 6x + 7. This simplification step provides the most straightforward form of the quadratic polynomial with the given zeroes. By setting k to 1, we eliminate the constant multiplier and arrive at a unique polynomial that satisfies the conditions. This simplified form is often preferred for its ease of use and interpretation. It directly showcases the relationship between the zeroes (3 + √2 and 3 - √2) and the coefficients of the polynomial. This final step highlights the importance of presenting mathematical results in their simplest form, enhancing clarity and facilitating further analysis or application.

Conclusion

In conclusion, forming a quadratic polynomial from its zeroes involves understanding the relationship between the zeroes and the coefficients of the polynomial. By calculating the sum and product of the zeroes, we can construct the polynomial using the general form k(x² - (α + β)x + αβ). In the case of zeroes 3 + √2 and 3 - √2, the quadratic polynomial is x² - 6x + 7 (when k = 1). This process not only provides a method for constructing quadratic polynomials but also reinforces fundamental algebraic concepts. Mastering these concepts is crucial for further studies in mathematics and related fields, as they form the building blocks for more advanced topics. The ability to construct polynomials from their roots is a testament to the elegance and interconnectedness of algebraic principles.