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Factor Theorem
Factor Theorem The Factor Theorem states that a polynomial of the form p(x) = (x - a)(x - b) ... (x - z) can be factored into a product of linear factors,...
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Factor Theorem The Factor Theorem states that a polynomial of the form p(x) = (x - a)(x - b) ... (x - z) can be factored into a product of linear factors,...
Factor Theorem
The Factor Theorem states that a polynomial of the form p(x) = (x - a)(x - b) ... (x - z) can be factored into a product of linear factors, where a, b, c, ... z are constants.
Proof:
The Factor Theorem can be proven using the following steps:
Assume the polynomial p(x) has factors of the form (x - a), (x - b), ..., (x - z).
Multiply these factors together.
The resulting expression must be equal to p(x).
Therefore, p(x) must be a product of linear factors.
Examples:
p(x) = x(x + 1) is a linear factor.
p(x) = (x - 2)(x + 3) is a quadratic factor.
p(x) = x^2 - 9 is a quadratic factor that cannot be factored into linear factors.
Applications of the Factor Theorem:
The Factor Theorem has many applications in mathematics, including:
Factoring polynomials for numerical values of a, b, c, ... z.
Solving linear equations and inequalities that involve polynomials.
Simplifying expressions that contain polynomials.
Conclusion:
The Factor Theorem is a powerful tool for understanding and manipulating polynomials. By understanding the Factor Theorem, students can gain a deeper understanding of the properties and applications of polynomials