The Factor Theorem states:
\[ f(x) = (x - c) \cdot q(x) + r \]
Where:
If \( f(c) = 0 \), then \( (x - c) \) is a factor of \( f(x) \).
The Factor Theorem is a mathematical theorem that provides a criterion for determining whether a given binomial is a factor of a given polynomial. It states that a polynomial \( f(x) \) has a factor \( (x - c) \) if and only if \( f(c) = 0 \). In other words, if a certain value, when substituted into the polynomial, results in zero, then the corresponding binomial is a factor of that polynomial. This theorem is a special case of the polynomial remainder theorem and is used in polynomial division and in finding the roots of a polynomial equation. It is a fundamental tool in algebra and calculus, and it simplifies the process of factoring polynomials and solving polynomial equations.
Let's assume the following polynomial:
Step 1: Substitute \( c = 1 \) into the polynomial:
\[ f(1) = 1^3 - 6 \cdot 1^2 + 11 \cdot 1 - 6 = 1 - 6 + 11 - 6 \]
Step 2: Simplify the expression:
\[ f(1) = 1 - 6 + 11 - 6 = 0 \]
Since \( f(1) = 0 \), the binomial \( (x - 1) \) is a factor of the polynomial \( x^3 - 6x^2 + 11x - 6 \).