Synthetic division

Definition

Synthetic division is a simplified method of dividing polynomials where only the coefficients are used. It is particularly useful for dividing by linear factors of the form $x - c$.

5 Must Know Facts For Your Next Test

1. Synthetic division can only be used when dividing by a linear polynomial in the form $x - c$.
2. It provides both the quotient and remainder of the division directly.
3. The process involves bringing down the leading coefficient, multiplying it by the root, and then adding it to the next coefficient iteratively.
4. Remainders in synthetic division can be interpreted as function values, specifically $f(c)$ where $c$ is the divisor's root.
5. Synthetic division is often used to test possible rational roots derived from the Rational Root Theorem.

Review Questions

• What type of polynomial must you be dividing by to use synthetic division?
• How does synthetic division help determine if a number is a root of a polynomial?
• What steps are involved in performing synthetic division?

Related terms

Polynomial Long Division: A traditional method for dividing polynomials, similar to long division with numbers but involving variables and coefficients.

Rational Root Theorem: A theorem that provides a possible list of rational roots for a polynomial equation based on its leading coefficient and constant term.

Factor Theorem: $c$ is a root of polynomial $f(x)$ if and only if $(x - c)$ is a factor of $f(x)$. Synthetic division helps verify this relationship.