(x-2i)(x%2B2i).jpeg "(x-4)(x-2i)(x+2i)")
Factoring and Expanding the Expression (x-4)(x-2i)(x+2i)
This expression involves complex numbers and can be expanded and factored in a few steps. Let's break it down.
Understanding the Expression
The expression (x-4)(x-2i)(x+2i) is a product of three factors:
- (x-4): A linear factor with a real root x=4.
- (x-2i): A linear factor with a complex root x=2i.
- (x+2i): A linear factor with a complex root x=-2i.
Expanding the Expression
To expand the expression, we can use the distributive property (or FOIL method) multiple times:
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Expand (x-2i)(x+2i):
(x-2i)(x+2i) = x² - (2i)² = x² + 4
Note: i² = -1
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Expand the result from step 1 with (x-4):
(x² + 4)(x-4) = x³ - 4x² + 4x - 16
Therefore, the expanded form of the expression is x³ - 4x² + 4x - 16.
Factoring the Expression
We can factor the expression back to its original form:
- Notice that the expression is a cubic polynomial: We can try to factor it by grouping or by using the Rational Root Theorem.
- The Rational Root Theorem helps us find possible rational roots: In this case, the possible rational roots are factors of -16 divided by factors of 1, which are ±1, ±2, ±4, ±8, ±16.
- We can test these possible roots by substituting them into the polynomial: We find that x=4 is a root.
- We can use synthetic division to divide the polynomial by (x-4): This gives us a quotient of x² + 4.
- We now have factored the polynomial as (x-4)(x² + 4):
- Finally, we recognize that x² + 4 is the difference of squares: (x² + 4) = (x+2i)(x-2i)
Therefore, the fully factored form of the expression is (x-4)(x-2i)(x+2i).
Key Points
- The expression (x-4)(x-2i)(x+2i) is a cubic polynomial with one real root (x=4) and two complex roots (x=2i, x=-2i).
- Complex roots always come in conjugate pairs.
- When expanding expressions with complex numbers, remember that i² = -1.
- Factoring and expanding expressions involving complex numbers can be done using the same techniques used for real numbers.