91Ó°ÊÓ

Understanding polynomial factorization is essential for working with polynomial equations. Factoring a polynomial means breaking it down into simpler components, or 'factors', that when multiplied together give you the original polynomial.
Let's consider the polynomial function from our exercise: \[ f(x) = x^4 + 2x^3 + 22x^2 + 50x - 75 \]
The goal is to express this polynomial in a product of simpler polynomial factors. For this, we often start with roots or zeros of the polynomial, which are the values of x where the polynomial equals zero. Once we identify these roots, we can use them to construct the factors.

The Rational Root Theorem is a helpful tool when we need to find possible rational roots (or zeros) of a polynomial function. According to this theorem, any possible rational root of a polynomial equation \[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \text{ ... } + a_1 x + a_0 \] must be a factor of the constant term \[ a_0 \] divided by a factor of the leading coefficient \[ a_n \].
For our polynomial, the constant term is -75, and the leading coefficient is 1. Potential rational roots are thus the factors of -75: \[ \text{±1, ±3, ±5, ±15, ±25, ±75} \] By testing these potential roots with synthetic division, we can confirm or reject them as actual roots of the polynomial.

Synthetic division is a streamlined way of dividing polynomials, making the process of root testing much faster. Here's how we use synthetic division to test the possible roots:
1. Use the root you are testing and place it on the left.
2. Write down the coefficients of the polynomial.
3. Bring the first coefficient straight down.
4. Multiply this number by the root and write the result under the next coefficient.
5. Add these numbers together and repeat until you go through all coefficients.
If the final result is zero, the number you tested is a root. For our polynomial, by testing -3 and finding the result to be zero, we confirmed that \[x + 3 \] is a factor. Repeating this process, we find that 1 is also a root, giving us another factor: \[x - 1 \].

The difference of squares is a useful method for further factorizing certain polynomials. This method applies when a polynomial can be expressed as \[ a^2 - b^2 \]. It factors into \[ (a - b)(a + b) \].
For our exercise, once we've reduced the polynomial to \[ x^2 + 25 \] through previous steps, we recognize that \[ x^2 + 25 \] can be seen in the form of a difference of squares if we think of it as \[ x^2 - (-25) \] or \[ x^2 - (5i)^2 \]. It factors further into the product of: \[ (x - 5i)(x + 5i) \].
Combining all factors, we get the fully factored form of the polynomial: \[ f(x) = (x + 3)(x - 1)(x - 5i)(x + 5i) \]. This shows all complex zeros effectively.