Polynomial Function With Given Roots: Find The Lowest Degree

Let's dive into the world of polynomials and figure out how to construct one with specific characteristics! This article will explore how to find the polynomial function with the lowest degree, given rational real coefficients, a leading coefficient of 3, and roots of √5 and 2. We'll break down the concepts step by step, making it easy to understand even if you're not a math whiz.

Understanding the Basics of Polynomial Functions

To kick things off, let's cover some foundational concepts related to polynomial functions. Polynomials are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A polynomial function can be expressed in the general form:

f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0

Where:

• a_n, a_{n-1}, ..., a_1, a_0 are the coefficients (constants).
• x is the variable.
• n is a non-negative integer representing the degree of the polynomial.

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial f(x) = 3x^4 + 2x^2 - x + 5, the degree is 4.

A root (or zero) of a polynomial function is a value of x that makes the function equal to zero, i.e., f(x) = 0. Roots are crucial for understanding the behavior and properties of polynomial functions. For instance, if a polynomial has a root r, then (x - r) is a factor of the polynomial.

In our problem, we are looking for a polynomial with specific roots and coefficients, so understanding these basics is essential.

Key Concepts: Rational Coefficients and Leading Coefficient

Before we jump into solving the problem, it's essential to understand two key concepts:

• Rational Coefficients: Rational coefficients are coefficients that can be expressed as a ratio of two integers (a fraction). For example, 1/2, -3, and 0 are rational numbers, while √2 and π are not.
• Leading Coefficient: The leading coefficient is the coefficient of the term with the highest power of the variable. In the polynomial f(x) = 3x^4 + 2x^2 - x + 5, the leading coefficient is 3.

In our problem, we are given that the polynomial must have rational real coefficients and a leading coefficient of 3. This information helps narrow down the possible solutions.

Finding the Polynomial Function

Now, let's tackle the problem head-on. We need to find the polynomial function of the lowest degree with rational real coefficients, a leading coefficient of 3, and roots √5 and 2. Here's how we can approach this:

Step 1: Understanding Conjugate Roots

Since the polynomial has rational real coefficients, and one of the roots is √5 (an irrational number), we must also consider its conjugate, -√5, as a root. This is because irrational roots of polynomials with rational coefficients always come in conjugate pairs. This concept is rooted in the Conjugate Root Theorem, which states that if a polynomial with real coefficients has a complex root a + bi, then its conjugate a - bi is also a root. In our case, √5 can be thought of as 0 + √5, so its conjugate is 0 - √5, or -√5.

Step 2: Constructing the Factors

Knowing the roots helps us construct the factors of the polynomial. If r is a root, then (x - r) is a factor. Therefore, since the roots are √5, -√5, and 2, the factors are:

• (x - √5)
• (x + √5)
• (x - 2)

Step 3: Multiplying the Factors

To find the polynomial, we multiply these factors together:

f(x) = (x - √5)(x + √5)(x - 2)

First, multiply the conjugate factors (x - √5) and (x + √5):

(x - √5)(x + √5) = x^2 - (√5)^2 = x^2 - 5

Now, multiply the result by the remaining factor (x - 2):

(x^2 - 5)(x - 2) = x^3 - 2x^2 - 5x + 10

Step 4: Adjusting the Leading Coefficient

The polynomial we have, x^3 - 2x^2 - 5x + 10, has a leading coefficient of 1. However, we need a polynomial with a leading coefficient of 3. To achieve this, we multiply the entire polynomial by 3:

f(x) = 3(x^3 - 2x^2 - 5x + 10) = 3x^3 - 6x^2 - 15x + 30

Step 5: Verifying the Solution

So, the polynomial function of the lowest degree with rational real coefficients, a leading coefficient of 3, and roots √5 and 2 is:

f(x) = 3x^3 - 6x^2 - 15x + 30

Let's verify that this polynomial indeed satisfies all the given conditions:

• Lowest Degree: The polynomial is of degree 3, which is the lowest possible degree for three roots (√5, -√5, and 2).
• Rational Real Coefficients: All the coefficients (3, -6, -15, and 30) are rational real numbers.
• Leading Coefficient of 3: The leading coefficient is indeed 3.
• Roots √5 and 2: We constructed the polynomial using these roots, so they are guaranteed to be roots of the polynomial.

Why the Conjugate Root Theorem Matters

The Conjugate Root Theorem is a cornerstone concept when dealing with polynomials that have rational coefficients. It ensures that if a polynomial has an irrational root of the form a + √b or a complex root of the form a + bi, its conjugate a - √b or a - bi must also be a root. This is because irrational and complex roots arise from quadratic factors with irrational or complex solutions, and these solutions always come in conjugate pairs when the coefficients are rational.

In our problem, if we had only considered the root √5 without including its conjugate -√5, we would not have arrived at a polynomial with rational coefficients. The inclusion of conjugate roots is essential for maintaining the rationality of the coefficients.

Practice Problems and Further Exploration

To solidify your understanding of polynomial functions and roots, let's consider a couple of practice problems:

1. Find the polynomial function of the lowest degree with rational real coefficients and roots 1 + i and 3.
2. What is the polynomial function with rational coefficients, a leading coefficient of 2, and roots -√2 and 1?

Exploring these problems will provide you with valuable practice in applying the concepts we've discussed. Additionally, consider delving deeper into related topics such as:

• The Fundamental Theorem of Algebra: This theorem states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. It's a foundational theorem in understanding polynomial equations.
• Descartes' Rule of Signs: This rule provides a method for determining the possible number of positive and negative real roots of a polynomial.
• Polynomial Division: Understanding how to divide polynomials is crucial for factoring and finding roots.

Conclusion

In conclusion, we've successfully navigated the process of finding the polynomial function of the lowest degree with specific characteristics. We started by understanding the basics of polynomial functions, including the concepts of rational coefficients, leading coefficients, and roots. Then, we applied the Conjugate Root Theorem to identify all necessary roots and constructed the polynomial by multiplying the corresponding factors. Finally, we verified that our solution met all the given conditions.

By grasping these concepts and practicing problem-solving, you'll be well-equipped to tackle a wide range of polynomial-related challenges. Keep exploring, keep practicing, and you'll master the world of polynomials in no time!

For further reading on polynomial functions, you can visit Khan Academy's Polynomial Arithmetic page. This resource offers a comprehensive overview of polynomial arithmetic and related concepts.