Remainder Theorem

Definition

The Remainder Theorem is a fundamental concept in polynomial division that allows for the determination of the remainder when a polynomial is divided by a linear expression. It provides a convenient way to evaluate the value of a polynomial at a specific point without having to perform the full polynomial division process.

5 Must Know Facts For Your Next Test

1. The Remainder Theorem states that if a polynomial $P(x)$ is divided by $(x - a)$, then the remainder is equal to $P(a)$.
2. The Remainder Theorem can be used to quickly evaluate the value of a polynomial at a specific point without performing the full division process.
3. The Remainder Theorem is closely related to the Factor Theorem, as it provides a way to determine if a linear expression $(x - a)$ is a factor of a polynomial $P(x)$.
4. Synthetic division is a convenient method for applying the Remainder Theorem, as it allows for the efficient division of a polynomial by a linear expression.
5. The Remainder Theorem is an important tool in the study of polynomial functions, as it enables the factorization and analysis of polynomials.

Review Questions

• Explain how the Remainder Theorem is used to evaluate the value of a polynomial at a specific point.
  • According to the Remainder Theorem, if a polynomial $P(x)$ is divided by $(x - a)$, then the remainder is equal to $P(a)$. This means that to evaluate the value of a polynomial $P(x)$ at a specific point $x = a$, you can simply plug in $a$ for $x$ in the original polynomial expression, without having to perform the full polynomial division process. This provides a convenient way to determine the value of a polynomial at a given point.
• Describe the relationship between the Remainder Theorem and the Factor Theorem.
  • The Remainder Theorem and the Factor Theorem are closely related. The Factor Theorem states that a linear expression $(x - a)$ is a factor of a polynomial $P(x)$ if and only if $P(a) = 0$. The Remainder Theorem builds on this by stating that if a polynomial $P(x)$ is divided by $(x - a)$, then the remainder is equal to $P(a)$. This means that if the remainder is zero, then $(x - a)$ is a factor of $P(x)$, as per the Factor Theorem. The Remainder Theorem provides a convenient way to determine if a linear expression is a factor of a polynomial.
• Explain how the Remainder Theorem can be used to factor polynomials.
  • The Remainder Theorem can be used as a tool for factoring polynomials. If we know that a linear expression $(x - a)$ is a factor of a polynomial $P(x)$, then we can use the Remainder Theorem to determine the value of $P(a)$. If $P(a) = 0$, then $(x - a)$ is a factor of $P(x)$, as per the Factor Theorem. By repeatedly applying the Remainder Theorem to find the roots of a polynomial, we can then use the factorization method to decompose the polynomial into its linear factors. This process of using the Remainder Theorem to factor polynomials is an important application of this key concept.

Related terms

Polynomial Division: The process of dividing one polynomial by another to obtain a quotient and a remainder.

Factor Theorem: A theorem that states that a linear expression $(x - a)$ is a factor of a polynomial $P(x)$ if and only if $P(a) = 0$.

Synthetic Division: A shortcut method for performing polynomial division, which involves the division of a polynomial by a linear expression of the form $(x - a)$.