Factor Theorem

Definition

The factor theorem is a fundamental concept in polynomial division that states that a polynomial $P(x)$ is divisible by $(x - a)$ if and only if $P(a) = 0$. In other words, the factor $(x - a)$ is a factor of the polynomial $P(x)$ if and only if the polynomial evaluates to zero when $x = a$.

5 Must Know Facts For Your Next Test

1. The factor theorem is a powerful tool for finding factors of polynomials and solving polynomial equations.
2. The factor theorem can be used to determine the roots of a polynomial equation by setting $P(x) = 0$ and solving for $x$.
3. The factor theorem is closely related to the remainder theorem, which provides a way to calculate the remainder when dividing a polynomial by a linear factor.
4. Polynomial factorization often involves applying the factor theorem to find the linear factors of a polynomial, which can then be used to find the remaining factors.
5. The factor theorem is a fundamental concept in the study of polynomial functions and their properties.

Review Questions

• Explain how the factor theorem can be used to determine the factors of a polynomial.
  • The factor theorem states that a polynomial $P(x)$ is divisible by $(x - a)$ if and only if $P(a) = 0$. This means that if you can find a value of $a$ that makes $P(a) = 0$, then $(x - a)$ is a factor of $P(x)$. By repeatedly applying the factor theorem to find the linear factors of a polynomial, you can then use polynomial factorization to express the polynomial as a product of its factors.
• Describe the relationship between the factor theorem and the remainder theorem.
  • The factor theorem and the remainder theorem are closely related. The remainder theorem states that if a polynomial $P(x)$ is divided by $(x - a)$, then the remainder is equal to $P(a)$. This means that if $P(a) = 0$, then $(x - a)$ is a factor of $P(x)$, which is the statement of the factor theorem. Conversely, if $(x - a)$ is a factor of $P(x)$, then $P(a) = 0$, which is the remainder theorem. The two theorems provide complementary ways of understanding the relationship between the factors of a polynomial and the values of the polynomial at those factors.
• Explain how the factor theorem can be used to solve polynomial equations.
  • To solve a polynomial equation $P(x) = 0$, you can apply the factor theorem to find the roots of the equation. The factor theorem states that $P(x)$ is divisible by $(x - a)$ if and only if $P(a) = 0$. This means that the values of $x$ that make $P(x) = 0$ are the roots of the polynomial equation. By finding the values of $a$ that satisfy $P(a) = 0$, you can determine the factors $(x - a)$ of the polynomial, and these factors represent the solutions to the polynomial equation.

Related terms

Polynomial Division: The process of dividing one polynomial by another, resulting in a quotient and a remainder.

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

Polynomial Factorization: The process of expressing a polynomial as a product of linear or irreducible quadratic factors.