Verified Answer :
- The Fundamental Theorem of Algebra confirms that a polynomial's degree is equal to its total number of complex roots.
- For \(f(x) = 3x^6 + 2x^5 + x^4 - 2x^3\), the highest power is 6, so there are \(\boxed{6}\) roots.
- For \(g(x) = 5x - 12x^2 + 3\), the highest power is 2, so there are \(\boxed{2}\) roots.
- For \(f(x) = (3x^4 + 1)^2\), the highest power is \(4 \times 2 = 8\), so there are \(\boxed{8}\) roots.
- For \(g(x) = (x - 5)^2 + 2x^3\), the expression simplifies to \(2x^3 + x^2 - 10x + 25\), giving a highest power of 3, so there are \(\boxed{3}\) roots.
Explanation
- Analysis of the Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that every non-zero, single-variable polynomial of degree \(n\) with complex coefficients has exactly \(n\) complex roots (when counted with multiplicity). To find the total number of roots, we simply need to determine the degree of each polynomial, which is the highest power of \(x\) present in the polynomial's expanded form.
- Analyzing the First Polynomial
For the polynomial \(f(x) = 3x^6 + 2x^5 + x^4 - 2x^3\), the term with the highest power of \(x\) is \(3x^6\). Therefore, the degree of this polynomial is 6, which means it has 6 complex roots.
- Analyzing the Second Polynomial
For the polynomial \(g(x) = 5x - 12x^2 + 3\), we rewrite it in standard form by rearranging the terms: \(g(x) = -12x^2 + 5x + 3\). The term with the highest power of \(x\) is \(-12x^2\), so the degree of this polynomial is 2. Consequently, it has 2 complex roots.
- Analyzing the Third Polynomial
For the polynomial \(f(x) = (3x^4 + 1)^2\), we apply the rule for the power of a power. Expanding this, we get \((3x^4)^2 + 2(3x^4)(1) + (1)^2 = 9x^8 + 6x^4 + 1\). The highest power of \(x\) is 8. Alternatively, we can use the rule that the degree of \((P(x))^k\) is \(k \times \text{degree}(P(x))\). Here, the degree of \((3x^4 + 1)\) is 4, and \(4 \times 2 = 8\). Thus, this polynomial has 8 complex roots.
- Analyzing the Fourth Polynomial
For the polynomial \(g(x) = (x - 5)^2 + 2x^3\), we first expand the squared term: \((x - 5)^2 = x^2 - 10x + 25\). Adding the remaining term, we have \(g(x) = x^2 - 10x + 25 + 2x^3\). Rearranging gives \(g(x) = 2x^3 + x^2 - 10x + 25\). The term with the highest power of \(x\) is \(2x^3\), so the degree of this polynomial is 3. This indicates it has 3 complex roots.
Examples
Determining the number of roots of a polynomial is a foundational skill in engineering and physics, especially when analyzing systems described by differential equations. For instance, in control theory, the stability of a physical system—like a drone's flight controller—often depends on the roots of its characteristic polynomial. Knowing the number of roots allows engineers to identify how many "modes of behavior" the system has, which is critical for ensuring the system behaves predictably and remains stable during operation.