91Ó°ÊÓ

Finding the zeros of a polynomial involves determining the values of the variable, typically represented as \( x \), that make the polynomial equal to zero. These values are also known as the roots of the polynomial.
For our polynomial \( P(x) = x^4 + 6x^2 + 9 \), we first need to express it in a form that allows us to solve it. By setting \( y = x^2 \), the polynomial becomes a quadratic: \( y^2 + 6y + 9 \).

Solving the quadratic equation \( y^2 + 6y + 9 = 0 \) by factoring, we get \( (y + 3)^2 = 0 \), which implies \( y = -3 \). Since \( y = x^2 \), we solve \( x^2 = -3 \), leading to zeros: \( x = i\sqrt{3} \) and \( x = -i\sqrt{3} \).
This involves complex numbers because the square root of a negative number leads to the imaginary unit \( i \).

Factoring a polynomial means breaking it down into simpler polynomials (of lower degrees), whose product results in the original polynomial. This is crucial as it plays a big role in solving polynomial equations and finding polynomial roots.
For \( P(x) = x^4 + 6x^2 + 9 \), once we determined the polynomial is a disguised quadratic in terms of \( y = x^2 \), it becomes straightforward to factor. We determined that \( (y + 3)^2 = 0 \) gives us \( y + 3 = 0 \) or \( y = -3 \).

Substituting back, \( x^2 = -3 \) leads us to factor the polynomial as \( (x^2 + 3)^2 \). Essentially, factoring involves realizing \((x^2 + 3) = (x - i\sqrt{3})(x + i\sqrt{3})\), creating the factorization \( P(x) = (x - i\sqrt{3})^2 (x + i\sqrt{3})^2 \). This approach utilizes complex numbers implicitly because of the need to incorporate \( i \) when solving for \( x \).

Complex numbers arise naturally when dealing with polynomial equations, especially when solutions extend beyond the real number domain. A complex number combines a real part and an imaginary part, and is typically written as \( a + bi \), where \( i \) is the imaginary unit, \( i^2 = -1 \).
When solving \( x^2 = -3 \), the result translates into complex solutions: \( x = \pm i\sqrt{3} \). The solutions are not just ordinary numbers but complex, as \( i \) accounts for the presence of negative under square roots.

Understanding complex numbers is critical for solving problems where real numbers do not suffice. In polynomial equations like \( P(x) = x^4 + 6x^2 + 9 \), complex roots ensure that all possible solutions are accounted for, preserving the fundamental theorem of algebra, which states every polynomial equation of degree \( n \) must have exactly \( n \) roots, considering multiplicity. In summary, complex numbers extend our computational horizon beyond the real number line.