(x-4-i)(x-4%2Bi).jpeg "(x-3)(x-4-i)(x-4+i)")
Factoring and Expanding Complex Polynomials: (x-3)(x-4-i)(x-4+i)
This article explores the process of factoring and expanding the polynomial expression: (x-3)(x-4-i)(x-4+i), where 'i' represents the imaginary unit (√-1).
Understanding Complex Numbers
Complex numbers are numbers that include both a real and an imaginary component. They are typically written in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. Complex numbers play a crucial role in various areas of mathematics and physics.
Factoring the Expression
The given expression is already in factored form, but we can simplify it by expanding it.
Expanding the Expression
To expand the expression, we can use the distributive property (also known as the FOIL method) multiple times.
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Start with the last two factors: (x - 4 - i)(x - 4 + i)
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Expand using FOIL:
- First: (x)(x) = x²
- Outer: (x)(+i) = +ix
- Inner: (-4)(x) = -4x
- Last: (-4)(+i) = -4i
- (-i)(x) = -ix
- (-i)(+i) = +i²
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Combine like terms: x² - 4x - 4x + ix - ix - 4i + 4i + i²
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Simplify using i² = -1: x² - 8x + 16 + (-1)
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Final simplified form: x² - 8x + 15
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Now we have: (x - 3)(x² - 8x + 15)
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Finally, expand the whole expression using distributive property:
- (x)(x²) = x³
- (x)(-8x) = -8x²
- (x)(15) = 15x
- (-3)(x²) = -3x²
- (-3)(-8x) = +24x
- (-3)(15) = -45
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Combine like terms: x³ - 11x² + 39x - 45
Conclusion
The expanded form of the expression (x-3)(x-4-i)(x-4+i) is x³ - 11x² + 39x - 45. This process demonstrates how to work with complex numbers and factor and expand polynomial expressions involving them.