The Factor Theorem provides a straightforward method to find the factors of a polynomial. It states that a polynomial f(x) has a factor (x - a) if and only if f(a) = 0. In simpler terms, if you can find a value a that makes the polynomial equal to zero, then (x - a) is a factor.
Here's how to use the Factor Theorem to find factors:
- Identify Potential Factors: Start by finding possible values for a that could make f(a) = 0. These values are typically the factors of the constant term of the polynomial.
- Test Potential Factors: Substitute each potential factor a into the polynomial f(x). If f(a) = 0, then (x - a) is a factor of the polynomial.
- Repeat the Process: Continue testing potential factors until you've found all the linear factors of the polynomial.
Example:
Let's find the factors of the polynomial f(x) = x³ - 6x² + 11x - 6.
- Step 1: The constant term is -6. Its factors are ±1, ±2, ±3, and ±6.
- Step 2: Let's test a = 1:
- f(1) = 1³ - 6(1)² + 11(1) - 6 = 0. Therefore, (x - 1) is a factor.
- Step 3: Now we can use polynomial division or synthetic division to divide f(x) by (x - 1). This gives us the quotient x² - 5x + 6.
- Step 4: We can now factor the quadratic expression x² - 5x + 6 as (x - 2)(x - 3).
Therefore, the factors of f(x) = x³ - 6x² + 11x - 6 are (x - 1)(x - 2)(x - 3).
Practical Insights:
- The Factor Theorem is particularly useful for polynomials with integer coefficients.
- While the Factor Theorem helps identify linear factors, it doesn't directly provide all factors of a polynomial, especially those with higher degrees.
- You may need to use other techniques, such as polynomial division or synthetic division, after finding a factor using the Factor Theorem.
