Finding Factors Of Polynomial Functions With Roots 3 + √5 And -6

Understanding the relationship between the roots and factors of polynomial functions is a fundamental concept in algebra. When a polynomial has irrational roots, it introduces an interesting twist. This article delves into the process of identifying factors of a polynomial function given its roots, particularly when those roots involve irrational numbers. Specifically, we will explore the scenario where a polynomial function f(x) has roots 3 + √5 and -6, and determine what must be a factor of f(x).

Understanding Polynomial Roots and Factors

In the realm of polynomial functions, a root is a value of x that makes the function equal to zero. In other words, if f(a) = 0, then a is a root of the polynomial f(x). Factors and roots are intrinsically linked. If a is a root of f(x), then (x - a) is a factor of f(x). This principle forms the basis for constructing polynomials from their roots and vice versa.

The Conjugate Root Theorem and Irrational Roots

When dealing with irrational roots, a crucial theorem comes into play: the Conjugate Root Theorem. This theorem states that if a polynomial with rational coefficients has an irrational root of the form a + √b, where a and b are rational and √b is irrational, then its conjugate a - √b must also be a root of the polynomial. This theorem is vital because irrational roots often come in conjugate pairs in polynomials with rational coefficients. This is because when we construct a polynomial with irrational roots, the irrational parts need to cancel out to leave us with rational coefficients. This cancellation happens naturally when we include both the root and its conjugate as factors.

For our specific problem, one of the given roots is 3 + √5. According to the Conjugate Root Theorem, since 3 + √5 is a root, its conjugate, 3 - √5, must also be a root of the polynomial f(x). This understanding is key to identifying the factors of the polynomial.

Determining the Factors of f(x)

Given that the roots of f(x) are 3 + √5, 3 - √5 (the conjugate), and -6, we can determine the factors of the polynomial. Recall that if a is a root, then (x - a) is a factor. Applying this to our roots:

• For the root 3 + √5, the corresponding factor is (x - (3 + √5)) which simplifies to (x - 3 - √5).
• For the root 3 - √5, the corresponding factor is (x - (3 - √5)) which simplifies to (x - 3 + √5). This is the most important one for this question, it needs to be right.
• For the root -6, the corresponding factor is (x - (-6)) which simplifies to (x + 6).

Therefore, f(x) must have factors of (x - 3 - √5), (x - 3 + √5), and (x + 6). The question asks us to identify a factor of f(x) from the given options. Among the provided choices, option B, (x - (3 - √5)), which is equivalent to (x - 3 + √5), is indeed a factor of f(x). This makes option B the correct answer.

Constructing the Polynomial (Optional)

While not necessary to answer the question, constructing the polynomial from its factors can provide a deeper understanding of the relationship between roots and factors. To construct the polynomial, we multiply its factors:

f(x) = (x - 3 - √5)(x - 3 + √5)(x + 6)

First, let's multiply the factors containing the conjugate roots:

(x - 3 - √5)(x - 3 + √5) = (x - 3)² - (√5)² = x² - 6x + 9 - 5 = x² - 6x + 4

Now, multiply this result by the remaining factor (x + 6):

(x² - 6x + 4)(x + 6) = x³ + 6x² - 6x² - 36x + 4x + 24 = x³ - 32x + 24

Thus, the polynomial f(x) can be expressed as x³ - 32x + 24. This confirms that the roots 3 + √5, 3 - √5, and -6 are indeed the roots of this polynomial.

Why the Other Options Are Incorrect

Let's briefly examine why the other options are incorrect:

• Option A: (x + (3 - √5)) This option represents a factor corresponding to the root -(3 - √5) = -3 + √5, which is not a root of f(x).
• Option C: (x + (5 + √3)) This option represents a factor corresponding to the root -(5 + √3), which is also not a root of f(x). Moreover, the presence of √3 suggests that the polynomial would have different irrational roots, which contradicts the given information.

Conclusion

In summary, when a polynomial function has roots of the form a + √b, its conjugate a - √b must also be a root, according to the Conjugate Root Theorem. Given the roots 3 + √5 and -6, we identified the factors of f(x) as (x - 3 + √5), (x - 3 - √5), and (x + 6). Therefore, option B, (x - (3 - √5)), is the correct answer. This exercise highlights the crucial connection between roots, factors, and the Conjugate Root Theorem in the study of polynomial functions. Understanding these concepts is essential for solving a wide range of algebraic problems and for a deeper appreciation of the structure of polynomial equations. Through this exploration, we've not only identified a factor of the given polynomial but also reinforced the fundamental principles that govern the behavior of polynomial functions with irrational roots. This knowledge serves as a cornerstone for tackling more complex problems in algebra and calculus, where the interplay between roots, factors, and coefficients becomes even more critical.