Finding the factors of a cubic equation (an equation of the form ax³ + bx² + cx + d = 0) typically involves these steps: finding a root and then using that root to simplify the equation into a quadratic, which is then easier to factor.
Here's a breakdown of the process:
1. Finding a Root (Using the Rational Root Theorem)
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The Rational Root Theorem: This theorem helps you identify potential rational roots (roots that can be expressed as a fraction p/q, where p and q are integers). The theorem states that if a rational number p/q is a root of the polynomial ax³ + bx² + cx + d = 0 (where a, b, c, and d are integer coefficients), then p must be a factor of the constant term d, and q must be a factor of the leading coefficient a.
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List Potential Roots: Identify all the factors of d (the constant term) and a (the leading coefficient). Create a list of all possible values of ±(factor of d) / (factor of a).
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Test Potential Roots: Substitute each potential root into the cubic equation. If the equation equals zero, you've found a root. The easiest way to test these roots is often using synthetic division.
Example: Consider the cubic equation x³ - 6x² + 11x - 6 = 0.
- a = 1 (leading coefficient)
- d = -6 (constant term)
Factors of d (-6): ±1, ±2, ±3, ±6
Factors of a (1): ±1
Possible rational roots: ±1, ±2, ±3, ±6
Let's test x = 1: (1)³ - 6(1)² + 11(1) - 6 = 1 - 6 + 11 - 6 = 0. Therefore, x = 1 is a root.
2. Finding a Linear Factor
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If 'a' is a root of the cubic equation, then (x - a) is a factor.
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Example (Continuing from above): Since x = 1 is a root, (x - 1) is a factor of x³ - 6x² + 11x - 6.
3. Finding the Quadratic Factor
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Polynomial Division (or Synthetic Division): Divide the original cubic equation by the linear factor you found. This will result in a quadratic equation.
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Example: Divide x³ - 6x² + 11x - 6 by (x - 1). The result is x² - 5x + 6.
4. Factoring the Quadratic Factor
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Factor the resulting quadratic equation (x² - 5x + 6 in our example). You can use factoring techniques, completing the square, or the quadratic formula.
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Example: x² - 5x + 6 factors to (x - 2)(x - 3).
5. Complete Factorization
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Combine the linear factor and the factored quadratic factor to get the complete factorization of the cubic equation.
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Example: The complete factorization of x³ - 6x² + 11x - 6 is (x - 1)(x - 2)(x - 3).
Summary:
- Use the Rational Root Theorem to find a possible root.
- Test the possible roots until you find one that works.
- Form a linear factor from the root you found.
- Divide the cubic polynomial by the linear factor to get a quadratic.
- Factor the quadratic.
- Combine the linear and quadratic factors.
