(x-4i)(x%2B4i).jpeg "(x-6)(x-4i)(x+4i)")
Expanding and Simplifying the Polynomial: (x - 6)(x - 4i)(x + 4i)
This expression represents a polynomial in factored form. To understand its expanded form and its properties, we need to multiply the factors together.
Expanding the Complex Factors
First, we focus on the complex factors: (x - 4i) and (x + 4i). This is a classic example of the difference of squares pattern. Remember: (a - b)(a + b) = a² - b²
Applying this pattern:
(x - 4i)(x + 4i) = x² - (4i)²
Since i² = -1, we have:
x² - (4i)² = x² - 16(-1) = x² + 16
Expanding the Full Polynomial
Now we multiply this result by the remaining factor (x - 6):
(x - 6)(x² + 16) = x(x² + 16) - 6(x² + 16)
Expanding further:
= x³ + 16x - 6x² - 96
Final Form
Rearranging terms in descending order of powers, we get the expanded form:
(x - 6)(x - 4i)(x + 4i) = x³ - 6x² + 16x - 96
Key Observations:
- Real Coefficients: Even though the original expression contained complex numbers, the expanded form has only real coefficients. This is a common outcome when working with complex conjugate pairs.
- Cubic Polynomial: The expanded polynomial is a cubic function, meaning its highest power is 3. This tells us it has a maximum of three roots (solutions).
This expanded form can be used for further analysis, such as finding the roots of the polynomial, graphing the function, or examining its behavior.