Polynomial Functions With 11 Roots Explained Fundamental Theorem Of Algebra

The Fundamental Theorem of Algebra is a cornerstone of polynomial theory, stating that every non-constant single-variable polynomial with complex coefficients has at least one complex root. A crucial corollary of this theorem extends its power, asserting that a polynomial of degree n, where n is a positive integer, has exactly n complex roots, counted with multiplicity. This means that a root appearing k times is counted as k roots. Understanding this theorem is paramount to correctly identifying the polynomial function possessing precisely 11 roots from the given options.

To determine which polynomial has exactly 11 roots, we need to analyze the degree of each polynomial. The degree of a polynomial is the highest power of the variable in the polynomial. According to the Fundamental Theorem of Algebra, a polynomial of degree n has n roots (counting multiplicities). Therefore, we are looking for a polynomial with a degree of 11.

Let's examine each option:

Analyzing Option A: f(x) = (x - 1)(x + 1)^11

In option A, understanding polynomial roots and their multiplicities is key. The function is given by f(x) = (x - 1)(x + 1)^11. This polynomial is already factored, making it easier to determine its roots and their multiplicities. The factor (x - 1) has a degree of 1, indicating a single root at x = 1. The factor (x + 1)^11 has a degree of 11, indicating a root at x = -1 with a multiplicity of 11. To find the total number of roots, we sum the multiplicities of all roots. In this case, we have 1 root from the factor (x - 1) and 11 roots from the factor (x + 1)^11, totaling 1 + 11 = 12 roots. Therefore, option A has 12 roots, not 11.

The degree of the polynomial in option A can also be determined by adding the exponents of the factors. The factor (x - 1) has a degree of 1, and the factor (x + 1)^11 has a degree of 11. The total degree of the polynomial is 1 + 11 = 12. According to the Fundamental Theorem of Algebra, this polynomial has 12 roots, which confirms our previous calculation.

Thus, option A does not have exactly 11 roots. The presence of the (x+1) term raised to the 11th power indicates 11 roots at x = -1, and the (x-1) term adds one more root at x = 1, bringing the total to 12 roots. It's crucial to consider the multiplicity of roots when determining the total number of roots for a polynomial. This understanding of polynomial root multiplicity is fundamental to solving problems related to the Fundamental Theorem of Algebra.

Dissecting Option B: f(x) = (x + 2)^3(x2 - 7x + 3)^4

Option B presents the function f(x) = (x + 2)^3(x2 - 7x + 3)^4. To calculate total roots using the Fundamental Theorem of Algebra, we need to determine the degree of this polynomial. The factor (x + 2)^3 has a degree of 3, indicating a root at x = -2 with a multiplicity of 3. The factor (x^2 - 7x + 3)^4 is a quadratic raised to the power of 4. A quadratic has a degree of 2, so raising it to the power of 4 multiplies the degree by 4, resulting in a degree of 2 * 4 = 8. This means the quadratic factor contributes 8 roots.

Summing the degrees of the factors, we have 3 (from (x + 2)^3) + 8 (from (x^2 - 7x + 3)^4) = 11. Therefore, the total degree of the polynomial is 11. According to the Fundamental Theorem of Algebra, this polynomial has 11 roots, counted with multiplicity. Hence, option B could be the correct answer. This careful polynomial degree analysis is essential for correctly applying the Fundamental Theorem of Algebra.

To further solidify our understanding, let's delve into how the roots are distributed. The term (x+2)^3 contributes three roots at x = -2. The term (x^2 - 7x + 3)^4 contributes eight roots, which are the roots of the quadratic x^2 - 7x + 3, each counted four times. The roots of the quadratic can be found using the quadratic formula, but for the purpose of this problem, it is sufficient to know that it contributes eight roots in total.

Evaluating Option C: f(x) = (x^5 + 7x + 14)^6

In option C, the function is given by f(x) = (x^5 + 7x + 14)^6. Here, determining polynomial degree requires us to analyze the expression inside the parentheses first. The term (x^5 + 7x + 14) is a polynomial of degree 5. This entire polynomial is then raised to the power of 6. When a polynomial is raised to a power, its degree is multiplied by that power. Therefore, the degree of the entire expression is 5 * 6 = 30. This means the polynomial f(x) has a degree of 30.

Based on the Fundamental Theorem of Algebra, a polynomial of degree 30 has 30 roots. Consequently, option C does not have exactly 11 roots. The key here is understanding degree multiplication rules for polynomials raised to a power. This concept is crucial for accurately determining the number of roots using the Fundamental Theorem of Algebra. Understanding the composite structure of the function is also vital; recognizing that the inner polynomial's degree is multiplied by the outer exponent allows for quick calculation of the overall degree.

Therefore, option C is incorrect because it has 30 roots, far exceeding the required 11 roots. The roots are the solutions to the equation x^5 + 7x + 14 = 0, each counted six times. While finding these roots explicitly might be complex, the Fundamental Theorem of Algebra provides a straightforward method to determine the total number of roots.

Assessing Option D: f(x) = 11

Option D presents a simple function: f(x) = 11. This is a constant polynomial. Constant polynomials have a degree of 0, as there is no variable x present. A polynomial of degree 0 has no roots, according to the Fundamental Theorem of Algebra. Therefore, option D does not have 11 roots. Understanding constant polynomial roots is a fundamental aspect of polynomial theory. Constant functions do not cross the x-axis, indicating the absence of real roots, and by extension, no complex roots either.

In this case, the function f(x) = 11 represents a horizontal line at y = 11. This line never intersects the x-axis, meaning there are no values of x for which f(x) = 0. Thus, it has zero roots. The Fundamental Theorem of Algebra applies to non-constant polynomials, and while a constant polynomial technically has a degree, it is considered to have zero roots in this context. This distinction is essential for accurately applying the theorem.

Conclusion: The Polynomial with Exactly 11 Roots

After carefully analyzing each option, we can confidently conclude that option B, f(x) = (x + 2)^3(x2 - 7x + 3)^4, is the polynomial function with exactly 11 roots. This determination is based on the Fundamental Theorem of Algebra, which states that a polynomial of degree n has n roots, counted with multiplicity. In option B, the polynomial has a degree of 11, confirming that it has 11 roots.

Understanding and applying the Fundamental Theorem of Algebra is crucial for solving problems related to polynomial roots. The process involves identifying the degree of the polynomial and considering the multiplicities of the roots. This comprehensive approach ensures accurate determination of the number of roots for any given polynomial function. Option B stood out due to its structure, where the sum of the exponents and degrees of each factor precisely matched the desired number of roots, making it the correct choice.