The factor theorem helps you determine the factors of a polynomial function by providing a direct link between the roots (or zeros) of the function and its linear factors. If you can find a value 'a' such that f(a) = 0, then (x - a) is a factor of f(x).
Here's a breakdown of how it works:
Understanding the Factor Theorem
The factor theorem is derived from the Remainder Theorem. It essentially states:
- For a polynomial function f(x), (x - a) is a factor of f(x) if and only if f(a) = 0. In other words, if plugging 'a' into the function results in zero, then (x-a) divides evenly into f(x).
How to Use the Factor Theorem to Find Factors
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Guess and Check (or use Rational Root Theorem): Start by guessing potential roots. Often, begin by testing factors of the constant term of the polynomial. The Rational Root Theorem provides a systematic way to generate a list of potential rational roots to test. This theorem states that if a polynomial has integer coefficients, then every rational root of the polynomial has the form p/q where p is a factor of the constant term and q is a factor of the leading coefficient.
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Evaluate the Function: For each potential root 'a', calculate f(a).
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Apply the Factor Theorem:
- If f(a) = 0, then (x - a) is a factor of f(x).
- If f(a) ≠ 0, then (x - a) is not a factor of f(x).
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Polynomial Division (or Synthetic Division): Once you've found a factor (x - a), divide the original polynomial f(x) by (x - a). The result will be a polynomial of a lower degree.
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Repeat: Repeat steps 1-4 with the new, lower-degree polynomial until you can factor it completely.
Example
Let's say you have the polynomial f(x) = x³ - 6x² + 11x - 6.
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Guess a root: Let's try a = 1.
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Evaluate: f(1) = (1)³ - 6(1)² + 11(1) - 6 = 1 - 6 + 11 - 6 = 0
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Apply the Factor Theorem: Since f(1) = 0, (x - 1) is a factor of f(x).
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Divide: Divide (x³ - 6x² + 11x - 6) by (x - 1). This gives you x² - 5x + 6.
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Factor the Result: Factor the quadratic x² - 5x + 6. This factors into (x - 2)(x - 3).
Therefore, the factors of x³ - 6x² + 11x - 6 are (x - 1), (x - 2), and (x - 3).
In Summary
The factor theorem provides a powerful tool for factoring polynomials. By finding values that make the polynomial equal to zero, you can identify corresponding linear factors and progressively break down the polynomial into simpler components.
