The factor theorem provides a method to determine if a linear expression (x - a) is a factor of a polynomial, f(x). This is based on evaluating the polynomial at x=a.
Understanding the Factor Theorem
According to the factor theorem:
- If f(a) = 0, then (x - a) is a factor of f(x). This means if you substitute 'a' into the polynomial and the result is zero, then the expression (x-a) divides the polynomial evenly, leaving no remainder.
- Conversely, if (x - a) is a factor of f(x), then f(a) = 0. This part of the theorem states that if we already know (x-a) is a factor, the result when we substitute 'a' into f(x) is zero.
Applying the Factor Theorem
Here's how to use the factor theorem to find factors:
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Identify potential factors: Look for possible values of 'a' that could make f(a) equal to zero. Often these are factors of the constant term of the polynomial, and you need to test them.
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Substitute and evaluate: Substitute the potential 'a' values into f(x). Calculate f(a) for each chosen value.
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Check for zeros: If f(a) = 0, then (x - a) is a factor.
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Repeat as needed: Continue this process to find all linear factors of the polynomial.
Example
Let's say we have the polynomial f(x) = x3 - 6x2 + 11x - 6. We want to find its factors.
- We'll try a = 1.
- f(1) = (1)3 - 6(1)2 + 11(1) - 6 = 1 - 6 + 11 - 6 = 0
- Since f(1) = 0, (x - 1) is a factor.
- Now let's try a = 2
- f(2) = (2)3 - 6(2)2 + 11(2) - 6 = 8 - 24 + 22 - 6 = 0
- Since f(2) = 0, (x - 2) is a factor.
- Let's try a = 3
- f(3) = (3)3 - 6(3)2 + 11(3) - 6 = 27 - 54 + 33 - 6 = 0
- Since f(3) = 0, (x - 3) is a factor.
Therefore, (x-1), (x-2), and (x-3) are all factors of the polynomial x3 - 6x2 + 11x - 6.
Summary
The factor theorem is a valuable tool for polynomial factorization. By testing potential 'a' values and seeing if f(a) equals zero, we can identify linear factors of the polynomial.
