Remainder Theorem

Definition

The Remainder Theorem states that the remainder of the division of a polynomial $f(x)$ by a linear divisor $(x - c)$ is $f(c)$. This theorem helps to quickly determine whether $c$ is a root of the polynomial.

5 Must Know Facts For Your Next Test

1. If $f(c) = 0$, then $(x - c)$ is a factor of the polynomial $f(x)$.
2. The Remainder Theorem simplifies finding remainders without performing long division.
3. It is particularly useful for evaluating polynomials at specific points.
4. The theorem can be extended to synthetic division, streamlining calculations further.
5. Understanding this theorem is essential for solving problems related to polynomial functions and their zeros.

Review Questions

• How does the Remainder Theorem help in determining if a value is a root of a polynomial?
• What is the remainder when $f(x) = x^3 - 4x^2 + 6x - 24$ is divided by $(x - 2)$?
• Explain how you would use the Remainder Theorem to find $f(3)$ for the polynomial $f(x) = 2x^4 - x^3 + 5x - 7$.

Related terms

Synthetic Division: A shortcut method for dividing a polynomial by a binomial of the form $(x - c)$ using only coefficients.

Factor Theorem: States that $(x - c)$ is a factor of the polynomial $f(x)$ if and only if $f(c) = 0$.

$Polynomial Function$: $A function that involves only non-negative integer powers of x.$