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According to the Fundamental Theorem of Algebra, which polynomial function has exactly 8 roots?


A$f(x)=\left(4 x^2-7\right)^3$
B$f(x)=\left(6 x^8-4 x^5-1\right)\left(3 x^2-4\right)$
C$f(x)=\left(3 x^2-4 x-5\right)\left(2 x^6-5\right)$
D$f(x)=\left(3 x^4+2 x\right)^4$

Verified Answer :

  • The Fundamental Theorem of Algebra states that a polynomial of degree \(n\) has exactly \(n\) roots.
  • We evaluate the degree of each given polynomial by adding the degrees of multiplied factors or multiplying the degree of a polynomial by its exponent.
  • The degrees are calculated as: 1) \(\deg = 6\), 2) \(\deg = 10\), 3) \(\deg = 8\), and 4) \(\deg = 16\).
  • Only the third polynomial has degree 8, meaning it has exactly 8 roots: \(\boxed{f(x) = (3x^2 - 4x - 5)(2x^6 - 5)}\).

Explanation

  1. Analyzing the Problem using the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a non-constant polynomial of degree \(n\) has exactly \(n\) roots in the complex number system, counted with their multiplicities. We are looking for a polynomial that has exactly 8 roots, which implies that the degree of the polynomial must be 8. We need to evaluate the degree for each of the given options:

  1. \(f(x) = (4x^2 - 7)^3\)
  2. \(f(x) = (6x^8 - 4x^5 - 1)(3x^2 - 4)\)
  3. \(f(x) = (3x^2 - 4x - 5)(2x^6 - 5)\)
  4. \(f(x) = (3x^4 + 2x)^4\)

The degree of a product of polynomials \(P(x) \cdot Q(x)\) is the sum of their degrees: \(\deg(P(x) \cdot Q(x)) = \deg(P(x)) + \deg(Q(x))\).
The degree of a power of a polynomial \(P(x)^n\) is the product of the degree of the polynomial and the exponent: \(\deg(P(x)^n) = n \cdot \deg(P(x))\).

  1. Calculating the Degree of Each Polynomial Function

Now we calculate the degree for each polynomial:

  1. For \(f(x) = (4x^2 - 7)^3\), the inner polynomial \(4x^2 - 7\) has a degree of 2. Applying the power rule for degrees, \(\deg(f) = 3 \cdot 2 = 6\).
  2. For \(f(x) = (6x^8 - 4x^5 - 1)(3x^2 - 4)\), the first factor \(6x^8 - 4x^5 - 1\) has degree 8, and the second factor \(3x^2 - 4\) has degree 2. Thus, \(\deg(f) = 8 + 2 = 10\).
  3. For \(f(x) = (3x^2 - 4x - 5)(2x^6 - 5)\), the first factor \(3x^2 - 4x - 5\) has degree 2, and the second factor \(2x^6 - 5\) has degree 6. Thus, \(\deg(f) = 2 + 6 = 8\).
  4. For \(f(x) = (3x^4 + 2x)^4\), the inner polynomial \(3x^4 + 2x\) has degree 4. Applying the power rule for degrees, \(\deg(f) = 4 \cdot 4 = 16\).
  1. Conclusion and Selection of the Correct Polynomial

Comparing the calculated degrees with the required number of roots (8):

  1. \(\deg(f) = 6 eq 8\)
  2. \(\deg(f) = 10 eq 8\)
  3. \(\deg(f) = 8\), which matches the requirement.
  4. \(\deg(f) = 16 eq 8\)

Therefore, the third function, \(f(x) = (3x^2 - 4x - 5)(2x^6 - 5)\), is the polynomial that has exactly 8 roots.

Examples

Understanding the degree of a polynomial is essential in fields like control theory and signal processing. For example, when designing a control system, the degree of the transfer function's denominator (the characteristic polynomial) determines the number of system poles. Knowing the number of poles allows engineers to analyze system stability and performance. Identifying the number of roots of a polynomial directly informs how a system will behave over time, helping to prevent instability or oscillations in mechanical or electrical engineering projects.

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