Remainder

Definition

The remainder is the part of a division operation that is left after dividing one polynomial by another. It represents what is left over when the divisor does not evenly divide the dividend.

5 Must Know Facts For Your Next Test

1. When dividing polynomials using long division, the degree of the remainder must be less than the degree of the divisor.
2. In synthetic division, the remainder appears at the bottom right of the final row in the synthetic division table.
3. If a polynomial $P(x)$ is divided by $(x - c)$ and $c$ is a root of $P(x)$, then the remainder will be zero.
4. The Remainder Theorem states that if a polynomial $P(x)$ is divided by $(x - c)$, the remainder is $P(c)$.
5. A non-zero remainder indicates that the divisor does not perfectly divide into the dividend.

Review Questions

• What does it mean if the remainder is zero when dividing two polynomials?
• Explain how to identify and interpret the remainder in synthetic division.
• According to the Remainder Theorem, what would be the remainder if $P(x) = x^3 - 4x + 2$ is divided by $(x - 1)$?

Related terms

Polynomial Long Division: A method for dividing a polynomial by another polynomial of lower degree, similar to long division with numbers.

Synthetic Division: A simplified form of polynomial division used when dividing by a linear factor.

Remainder Theorem: \text{For any polynomial } P(x), \text{the remainder of } P(x) \text{ divided by } (x-c) \text{is} P(c).