Factor Theorem

Definition

The Factor Theorem is a fundamental principle in polynomial factorization. It 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 0 when $x$ is replaced by $a$.

5 Must Know Facts For Your Next Test

1. The Factor Theorem provides a way to test whether a given expression is a factor of a polynomial.
2. If $P(a) = 0$, then $(x-a)$ is a factor of $P(x)$.
3. The Factor Theorem can be used to find the roots of a polynomial equation by setting $P(x) = 0$ and solving for $x$.
4. Factorization of polynomials is an important technique in solving polynomial equations and simplifying algebraic expressions.
5. The Factor Theorem is closely related to the Remainder Theorem, which provides a way to find the remainder when a polynomial is divided by a linear expression.

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 we can find a value of $a$ for which $P(a) = 0$, then $(x-a)$ is a factor of $P(x)$. We can use this principle to systematically test different values of $a$ and identify the factors of the polynomial.
• 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 $P(x)$, as stated by the Factor Theorem. Conversely, if $(x-a)$ is a factor of $P(x)$, then $P(a) = 0$, as per the Factor Theorem. The two theorems provide complementary ways to analyze the factorization of polynomials.
• Explain how the Factor Theorem can be used to solve polynomial equations.
  • The Factor Theorem can be used to solve polynomial equations by setting the polynomial expression equal to zero and then finding the values of $x$ that satisfy the equation. If $P(a) = 0$, then $(x-a)$ is a factor of $P(x)$, and $a$ is a root of the polynomial equation $P(x) = 0$. By repeatedly applying the Factor Theorem to find the roots of the polynomial, one can factor the polynomial and solve the equation.

Related terms

Polynomial:

A polynomial is an expression consisting of variables and coefficients, where the variables are only raised to non-negative integer powers.

Factorization:

Factorization is the process of expressing a polynomial as a product of smaller polynomials, known as factors.

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