Factor Theorem

Definition

The Factor Theorem is a fundamental principle in polynomial algebra that establishes a relationship between the factors of a polynomial and its zeros. It provides a way to determine whether a given expression is a factor of a polynomial and to find the roots or zeros of a polynomial function.

5 Must Know Facts For Your Next Test

1. The Factor Theorem states that a polynomial $P(x)$ is divisible by $(x - a)$ if and only if $P(a) = 0$.
2. The Factor Theorem can be used to find the zeros or roots of a polynomial function by setting the polynomial equal to zero and solving for the values of $x$ that satisfy the equation.
3. If a polynomial $P(x)$ has a factor $(x - a)$, then $a$ is a zero or root of the polynomial function $P(x)$.
4. The Factor Theorem can be used to simplify the division of polynomials by factoring the divisor and applying the Remainder Theorem.
5. The Factor Theorem is closely related to the Fundamental Theorem of Algebra, which states that every non-constant polynomial function has at least one complex root.

Review Questions

• Explain how the Factor Theorem can be used to find the zeros or roots of a polynomial function.
  • According to the Factor Theorem, a polynomial $P(x)$ is divisible by $(x - a)$ if and only if $P(a) = 0$. This means that the values of $x$ that satisfy the equation $P(x) = 0$ are the zeros or roots of the polynomial function. By setting the polynomial equal to zero and solving for the values of $x$, you can determine the roots of the polynomial function using the Factor Theorem.
• Describe how the Factor Theorem can be used to simplify the division of polynomials.
  • The Factor Theorem states that if a polynomial $P(x)$ is divided by $(x - a)$, then the remainder is $P(a)$. This means that if you can factor the divisor $(x - a)$ from the polynomial $P(x)$, you can apply the Remainder Theorem to simplify the division process. By finding the factors of the divisor and evaluating the polynomial at the corresponding values of $x$, you can determine the remainder without having to perform the full division algorithm.
• Analyze the relationship between the Factor Theorem and the Fundamental Theorem of Algebra.
  • The Factor Theorem and the Fundamental Theorem of Algebra are closely related in the context of polynomial functions. The Fundamental Theorem of Algebra states that every non-constant polynomial function has at least one complex root. This means that every polynomial can be factored into a product of linear factors of the form $(x - a)$, where $a$ is a root of the polynomial. The Factor Theorem, in turn, establishes the connection between these linear factors and the zeros or roots of the polynomial function. Together, these theorems provide a powerful framework for understanding the properties and behavior of polynomial functions.

Related terms

Polynomial Function: A polynomial function is an algebraic function that can be written as the sum of a finite number of non-negative integer powers of the independent variable, with coefficients.

Divisibility: Divisibility refers to the property of one number being evenly divisible by another number, without a remainder.

Remainder Theorem: The Remainder Theorem states that if a polynomial $P(x)$ is divided by $(x - a)$, then the remainder is $P(a)$.