Remainder

Definition

The remainder is the amount left over after dividing one number or expression by another. It represents the part of the dividend that is not evenly divisible by the divisor, providing information about the relationship between the two quantities.

5 Must Know Facts For Your Next Test

1. The remainder provides information about the relationship between the dividend and divisor, indicating how much of the dividend is not evenly divisible by the divisor.
2. When dividing polynomials, the remainder represents the part of the dividend polynomial that is not divisible by the divisor polynomial.
3. The remainder theorem states that if a polynomial $P(x)$ is divided by $(x - a)$, then the remainder is $P(a)$.
4. The factor theorem states that if the remainder of dividing a polynomial $P(x)$ by $(x - a)$ is 0, then $(x - a)$ is a factor of $P(x)$.
5. The degree of the remainder polynomial will always be less than the degree of the divisor polynomial when dividing polynomials.

Review Questions

• Explain the relationship between the remainder and the dividend and divisor in a division operation.
  • The remainder represents the part of the dividend that is not evenly divisible by the divisor. It provides information about the relationship between the two quantities, indicating how much of the dividend is left over after the division process. The remainder will always be less than the divisor and will have the same unit as the dividend.
• Describe the role of the remainder in the context of dividing polynomials.
  • When dividing polynomials, the remainder represents the part of the dividend polynomial that is not divisible by the divisor polynomial. The remainder theorem states that if a polynomial $P(x)$ is divided by $(x - a)$, then the remainder is $P(a)$. Additionally, the factor theorem states that if the remainder of dividing a polynomial $P(x)$ by $(x - a)$ is 0, then $(x - a)$ is a factor of $P(x)$. The degree of the remainder polynomial will always be less than the degree of the divisor polynomial.
• Analyze the significance of the remainder in the context of polynomial division and its applications in solving related problems.
  • The remainder in polynomial division is crucial because it provides valuable information about the relationship between the dividend and divisor polynomials. It can be used to determine factors of the dividend polynomial, as stated by the factor theorem. Additionally, the remainder theorem allows for the evaluation of a polynomial at a specific point, which can be useful in solving a variety of polynomial problems. Understanding the properties and significance of the remainder is essential for effectively dividing polynomials and applying the related concepts to solve more complex mathematical problems.

Related terms

Dividend: The number or expression that is being divided in a division operation.

Divisor: The number or expression by which the dividend is being divided.

Quotient: The result of dividing the dividend by the divisor, representing the number of times the divisor goes into the dividend.