Finding Roots And Verifying Relationships In Quadratic Polynomial 3x² - 2√6x + 2
In this comprehensive exploration, we will delve into the process of finding the zeros of the quadratic polynomial 3x² - 2√6x + 2 and meticulously verify the relationships between these zeros and the coefficients of the polynomial. This involves factoring the quadratic equation, identifying the roots, and then applying the established formulas that connect the zeros and coefficients. Our discussion will not only reinforce the fundamental concepts of quadratic equations but also highlight the elegant mathematical relationships that govern them.
Understanding Quadratic Polynomials and Their Zeros
Before we dive into the specifics of the given polynomial, it is crucial to have a firm grasp of what quadratic polynomials are and what their zeros represent. A quadratic polynomial is a polynomial of degree two, generally expressed in the form ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The zeros of a quadratic polynomial are the values of x for which the polynomial equals zero. These zeros are also known as the roots of the quadratic equation ax² + bx + c = 0. Geometrically, the zeros represent the points where the parabola defined by the quadratic polynomial intersects the x-axis.
Finding the zeros of a quadratic polynomial is a fundamental problem in algebra, and there are several methods to accomplish this, including factoring, completing the square, and using the quadratic formula. Each method has its strengths and is suitable for different types of quadratic polynomials. In this case, we will explore the method of factoring and demonstrate how it can be effectively used to find the zeros of the given polynomial.
Factoring Quadratic Polynomials
Factoring a quadratic polynomial involves expressing it as a product of two linear factors. This method is particularly efficient when the polynomial can be easily factored. For the quadratic polynomial 3x² - 2√6x + 2, we aim to find two binomials such that their product yields the original polynomial. This often involves identifying two numbers that multiply to give the constant term (c) and add up to give the coefficient of the linear term (b).
In our given polynomial, the coefficients involve a radical term, which might initially appear daunting. However, with careful observation and application of algebraic principles, we can effectively factor it. The key is to recognize that the middle term involves √6, which suggests that the factors might also involve √6. This insight helps narrow down the possible factors and simplifies the factoring process.
Relationship Between Zeros and Coefficients
A crucial aspect of quadratic polynomials is the relationship between their zeros and coefficients. For a quadratic polynomial ax² + bx + c, if α and β are the zeros, then the following relationships hold:
- Sum of zeros: α + β = -b/a
- Product of zeros: αβ = c/a
These relationships provide a powerful tool for verifying the correctness of the zeros we find and for understanding the structure of quadratic equations. They connect the roots of the equation to its coefficients, offering a deeper insight into the nature of quadratic polynomials.
Finding the Zeros of 3x² - 2√6x + 2
Now, let's apply our understanding to find the zeros of the quadratic polynomial 3x² - 2√6x + 2. We will use the factoring method, as it is particularly well-suited for this polynomial.
Step 1: Factoring the Polynomial
To factor the polynomial, we need to express it as a product of two binomials. We look for two numbers that multiply to give the product of the leading coefficient (3) and the constant term (2), which is 6, and add up to the coefficient of the linear term (-2√6). These two numbers are -√6 and -√6, since (-√6) * (-√6) = 6 and (-√6) + (-√6) = -2√6.
Using these numbers, we can rewrite the middle term of the polynomial and factor by grouping:
3x² - 2√6x + 2 = 3x² - √6x - √6x + 2
Now, we group the terms and factor out the common factors:
3x² - √6x - √6x + 2 = √3x(√3x - √2) - √2(√3x - √2)
We can see that (√3x - √2) is a common factor, so we factor it out:
√3x(√3x - √2) - √2(√3x - √2) = (√3x - √2)(√3x - √2)
Thus, the factored form of the polynomial is (√3x - √2)².
Step 2: Finding the Zeros
To find the zeros, we set the factored polynomial equal to zero and solve for x:
(√3x - √2)² = 0
Taking the square root of both sides, we get:
√3x - √2 = 0
Now, we solve for x:
√3x = √2
x = √2 / √3
To rationalize the denominator, we multiply the numerator and denominator by √3:
x = (√2 * √3) / (√3 * √3)
x = √6 / 3
Since the factor (√3x - √2) appears twice, we have a repeated root. Therefore, the polynomial has one distinct zero, x = √6 / 3, with a multiplicity of 2.
Verifying the Relationship Between Zeros and Coefficients
Having found the zero of the polynomial, we now proceed to verify the relationship between the zero and the coefficients. We will use the formulas for the sum and product of zeros.
Step 1: Identifying the Coefficients
In the quadratic polynomial 3x² - 2√6x + 2, the coefficients are:
- a = 3
- b = -2√6
- c = 2
Step 2: Applying the Sum of Zeros Formula
Since we have a repeated root, both zeros are α = √6 / 3 and β = √6 / 3. The sum of the zeros is:
α + β = (√6 / 3) + (√6 / 3) = 2√6 / 3
According to the formula, the sum of zeros should also be equal to -b/a. Let's verify this:
-b/a = -(-2√6) / 3 = 2√6 / 3
The sum of zeros calculated directly matches the value obtained using the formula, thus verifying the relationship.
Step 3: Applying the Product of Zeros Formula
The product of the zeros is:
αβ = (√6 / 3) * (√6 / 3) = 6 / 9 = 2 / 3
According to the formula, the product of zeros should also be equal to c/a. Let's verify this:
c/a = 2 / 3
The product of zeros calculated directly matches the value obtained using the formula, further verifying the relationship.
Conclusion
In this detailed exploration, we successfully found the zero of the quadratic polynomial 3x² - 2√6x + 2 using the factoring method. We identified the zero as x = √6 / 3 with a multiplicity of 2. Furthermore, we meticulously verified the relationships between the zero and the coefficients of the polynomial using the formulas for the sum and product of zeros. Our calculations confirmed that the sum of the zeros (α + β) is equal to -b/a and the product of the zeros (αβ) is equal to c/a, thus validating the fundamental relationships that govern quadratic polynomials. This exercise not only reinforces our understanding of quadratic equations but also demonstrates the elegant connections between the roots and coefficients, providing a deeper appreciation for the structure of algebraic expressions.
This comprehensive analysis underscores the importance of mastering the techniques for finding zeros and verifying relationships in quadratic polynomials, as these concepts are foundational to many areas of mathematics and its applications. By understanding these principles, students and practitioners can effectively solve a wide range of problems and gain a deeper insight into the mathematical world.
