(x-6)(x%2B7i)(x-7i).jpeg "(x-6)(x-6)(x+7i)(x-7i)")
Factoring and Simplifying Complex Expressions: (x-6)(x-6)(x+7i)(x-7i)
This expression involves both real and imaginary components, making it a bit more complex to work with. Here's a breakdown of how to factor and simplify it:
Understanding the Basics
- Real Numbers: Numbers that can be plotted on a number line (e.g., 2, -5, 0.75).
- Imaginary Numbers: Numbers involving the imaginary unit 'i', where i² = -1 (e.g., 3i, -2i).
- Complex Numbers: Numbers composed of a real part and an imaginary part (e.g., 2 + 3i, -5 - 2i).
Factoring the Expression
The expression (x-6)(x-6)(x+7i)(x-7i) is already mostly factored. Let's break it down:
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Repeated Factors: The first two factors (x-6)(x-6) represent a perfect square - (x-6)².
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Complex Conjugates: The last two factors (x+7i)(x-7i) are complex conjugates - they have the same real part but opposite imaginary parts.
Simplifying using the Difference of Squares
The key to simplifying the expression lies in recognizing the complex conjugates. The difference of squares pattern states:
(a + b)(a - b) = a² - b²
Applying this to our expression:
(x + 7i)(x - 7i) = x² - (7i)² = x² + 49
Final Result
Combining everything, the simplified expression is:
(x-6)²(x² + 49)
This is a quartic polynomial (degree 4) with both real and imaginary components.
Important Note: While the expression can be factored, it cannot be further simplified into linear factors (factors of degree 1) using only real numbers. This is because the complex conjugate factors contribute to the polynomial's irreducible nature over real numbers.