(x-4i)(x%2B4i).jpeg "(x-3)(x-4i)(x+4i)")
Factoring and Expanding Complex Polynomials: (x-3)(x-4i)(x+4i)
This article explores the process of factoring and expanding the polynomial expression (x-3)(x-4i)(x+4i). We will utilize the properties of complex numbers to simplify the expression and arrive at a standard polynomial form.
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as the square root of -1 (i² = -1).
Factoring the Expression
The given expression contains a pair of complex conjugates: (x-4i) and (x+4i). A key property of complex conjugates is that their product results in a real number.
Expanding (x-4i)(x+4i)
(x-4i)(x+4i) = x² - (4i)² = x² - 16i² = x² + 16 (since i² = -1)
Substituting the Product
Now, we can substitute this result back into the original expression:
(x-3)(x-4i)(x+4i) = (x-3)(x² + 16)
Expanding the Final Expression
Finally, we expand the remaining product to obtain the polynomial in standard form:
(x-3)(x² + 16) = x³ + 16x - 3x² - 48 = x³ - 3x² + 16x - 48
Conclusion
By understanding the properties of complex numbers and complex conjugates, we can simplify and expand the expression (x-3)(x-4i)(x+4i) to arrive at the polynomial x³ - 3x² + 16x - 48. This process demonstrates how to manipulate complex polynomial expressions to obtain a standard polynomial form.