Verified Answer :
- Calculate the degree of each polynomial by summing the multiplicities (exponents) of its linear factors.
- For \(f(x) = (x+1)(x-3)(x-4)\), the total number of roots is \(1+1+1 = \boxed{3}\).
- For \(f(x) = (x-6)^2(x+2)^2\), the total number of roots is \(2+2 = \boxed{4}\).
- For \(f(x) = (x+5)^3(x-9)(x+1)\), the total number of roots is \(3+1+1 = \boxed{5}\).
- For \(f(x) = (x+2)(x-1)[x-(4+3i)][x-(4-3i)]\), the total number of roots is \(1+1+1+1 = \boxed{4}\).
Explanation
- Analyzing the Degree of a Polynomial
The degree of a polynomial expressed in factored form, \(f(x) = c(x-r_1)^{m_1}(x-r_2)^{m_2}\dots(x-r_k)^{m_k}\), is the sum of the multiplicities \(m_1 + m_2 + \dots + m_k\). According to the Fundamental Theorem of Algebra, a polynomial of degree \(n\) has exactly \(n\) roots when counting their multiplicities. We will analyze each polynomial to determine the total number of roots by summing the exponents of its linear factors.
- Polynomial 1: \(f(x) = (x+1)(x-3)(x-4)\)
For the first polynomial \(f(x) = (x+1)(x-3)(x-4)\), each factor is implicitly raised to the power of 1. The sum of the multiplicities is \(1 + 1 + 1 = 3\). Therefore, the total number of roots for this polynomial is 3.
- Polynomial 2: \(f(x) = (x-6)^2(x+2)^2\)
For the second polynomial \(f(x) = (x-6)^2(x+2)^2\), we identify the factors \((x-6)\) with multiplicity 2 and \((x+2)\) with multiplicity 2. The sum of the multiplicities is \(2 + 2 = 4\). Therefore, the total number of roots for this polynomial is 4.
- Polynomial 3: \(f(x) = (x+5)^3(x-9)(x+1)\)
For the third polynomial \(f(x) = (x+5)^3(x-9)(x+1)\), we identify the factors \((x+5)\) with multiplicity 3, \((x-9)\) with multiplicity 1, and \((x+1)\) with multiplicity 1. The sum of the multiplicities is \(3 + 1 + 1 = 5\). Therefore, the total number of roots for this polynomial is 5.
- Polynomial 4: \(f(x) = (x+2)(x-1)[x-(4+3i)][x-(4-3i)]\)
For the fourth polynomial \(f(x) = (x+2)(x-1)[x-(4+3i)][x-(4-3i)]\), we identify four linear factors: \((x+2)\), \((x-1)\), \([x-(4+3i)]\), and \([x-(4-3i)]\). Each factor has a multiplicity of 1. The sum of the multiplicities is \(1 + 1 + 1 + 1 = 4\). Therefore, the total number of roots for this polynomial is 4.
Examples
Understanding the number of roots is crucial in engineering and physics, particularly when analyzing the stability of a system. For example, in control systems theory, a system's response is often determined by the roots (poles) of its transfer function. By representing a system's behavior through a polynomial, engineers can immediately identify how many modes of response the system has by simply looking at the degree, which is determined by the total count of roots, including complex and repeated ones. This allows them to predict whether the system will oscillate or remain stable over time.