91Ó°ÊÓ

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. In our exercise, the given roots are \( \sqrt{3} \), \( -\sqrt{3} \), \( 4i \), and \( -4i \) with specific multiplicities. Roots can be real or complex numbers and are fundamental to shaping the polynomial's structure.
Multiple roots with the same value indicate that these roots affect the polynomial multiple times, as denoted by their multiplicity. For example, if \( \sqrt{3} \) is a root with multiplicity 2, it appears twice in the factorization of the polynomial. This affects the polynomial's behavior and factorization.
Understanding the multiplicity of roots helps in constructing the polynomial and gives insight into the graph of the polynomial, such as whether it will touch or cross the x-axis at the root point.

Factored form refers to expressing a polynomial as a product of its factors. Writing a polynomial in its factored form reveals its roots easily, as each factor corresponds to a root of the polynomial. In our problem, each root contributes a factor.

• Real Roots: For \( \sqrt{3} \) and \( -\sqrt{3} \), the factors are \( (x - \sqrt{3}) \) and \( (x + \sqrt{3}) \).
• Complex Roots: For \( 4i \) and \( -4i \) the factors are \( (x - 4i) \) and \( (x + 4i) \).

By multiplying these factors together, considering the multiplicity of each root, we achieve the polynomial's factored form. The real beauty of the factored form is it clearly displays the roots, which are the solutions to the polynomial equation \( f(x) = 0 \).

Complex numbers extend our number system beyond real numbers. They are composed of a real part and an imaginary part. In this exercise, \( 4i \) and \( -4i \) are complex roots, where "i" is the imaginary unit representing \( \sqrt{-1} \).
When a polynomial has complex roots, they come in conjugate pairs if the coefficients of the polynomial are real numbers. This ensures the polynomial evaluates to real values for real inputs. So, when we have the complex roots \( 4i \) and \( -4i \), they form a pair that can be multiplied:
\((x - 4i)(x + 4i) = x^2 + 16.\)
This identity transforms the complex conjugate pair into a quadratic expression with real coefficients. Thus, complex numbers allow us to find solutions to polynomials that wouldn't be possible with only real numbers.

Quadratic identities are used to simplify expressions and solve equations involving squared terms. Two important identities we used in this problem include the difference of squares and its application in the context of complex roots.
The difference of squares states that \((a-b)(a+b) = a^2 - b^2\). This identity allows us to handle both real and complex roots efficiently.
For example, to incorporate \(\sqrt{3}\) and \(-\sqrt{3} \) as roots, we use \((x - \sqrt{3})(x + \sqrt{3}) = x^2 - 3\). This identity simplifies the expression into a manageable quadratic with real coefficients.
Similarly, with complex roots \(x - 4i\) and \(x + 4i\), applying this identity yields \(x^2 + 16\). Quadratic identities are thus crucial to expressing polynomials in a simplifyed manner, to easily identify roots and solve polynomial equations.