(x-3-i)(x-3%2Bi).jpeg "(x-2)(x-3-i)(x-3+i)")
Factoring and Solving Cubic Equations: A Comprehensive Guide
This article explores the factorization and solution of the cubic equation represented by the expression:
(x - 2)(x - 3 - i)(x - 3 + i)
Understanding the Expression
This expression represents a cubic equation, meaning it has a highest power of x as 3. The equation is already factored into three linear factors:
- (x - 2): This factor indicates a root of the equation at x = 2.
- (x - 3 - i): This factor represents a complex root of the equation at x = 3 + i, where 'i' is the imaginary unit (√-1).
- (x - 3 + i): This factor represents another complex root, which is the conjugate of the previous one. Complex roots always come in conjugate pairs.
Expanding the Expression
To find the original cubic equation, we need to expand the factored expression:
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Expand the last two factors:
(x - 3 - i)(x - 3 + i) = x² - 3x + ix - 3x + 9 - 3i - ix + 3i - i²
Simplifying this, we get:
x² - 6x + 10
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Multiply the result with the first factor:
(x - 2)(x² - 6x + 10) = x³ - 6x² + 10x - 2x² + 12x - 20
Finally, we get the expanded cubic equation:
x³ - 8x² + 22x - 20
Finding the Roots
We already know the roots of the equation from the factored expression:
- x = 2
- x = 3 + i
- x = 3 - i
This confirms that the equation has one real root (x = 2) and two complex roots (x = 3 + i, x = 3 - i).
Key Takeaways
- Complex roots always come in conjugate pairs: This is a fundamental property of polynomial equations with real coefficients.
- Factoring helps in finding roots: By factoring an equation, we can easily identify its roots.
- Expanding the factored expression gives the original equation: This demonstrates the relationship between factored and expanded forms of a polynomial.
This article provides a detailed explanation of factoring and solving a cubic equation, highlighting the importance of complex roots and their conjugate pairs.