key term - Divisible

Definition

Divisible is a mathematical term that describes a number or expression that can be evenly divided by another number without leaving a remainder. It is a fundamental concept in the context of dividing polynomials, as it determines the feasibility and ease of the division process.

5 Must Know Facts For Your Next Test

1. A polynomial $P(x)$ is divisible by $(x - a)$ if and only if $P(a) = 0$.
2. The degree of the divisor must be less than or equal to the degree of the dividend for the division to be possible.
3. When dividing polynomials, the divisor is the expression being divided, and the dividend is the expression being divided into.
4. The Remainder Theorem can be used to determine if a polynomial is divisible by a given linear expression.
5. Factoring a polynomial can help determine if it is divisible by a particular expression.

Review Questions

• Explain the relationship between the divisibility of a polynomial and the Remainder Theorem.
  • According to the Remainder Theorem, a polynomial $P(x)$ is divisible by $(x - a)$ if and only if $P(a) = 0$. This means that if a polynomial is divisible by a linear expression $(x - a)$, then the value of the polynomial when $x = a$ will be zero. Conversely, if the value of the polynomial at $x = a$ is zero, then the polynomial is divisible by $(x - a)$. This connection between divisibility and the Remainder Theorem is crucial in determining whether a polynomial is divisible by a given expression.
• Describe the role of the degree of the divisor in the polynomial division process.
  • The degree of the divisor is an important factor in determining the feasibility of polynomial division. For the division to be possible, the degree of the divisor must be less than or equal to the degree of the dividend. If the degree of the divisor is greater than the degree of the dividend, then the division cannot be performed, as there would be no way to match the leading terms of the dividend and the divisor. This requirement ensures that the division process can be carried out successfully, resulting in a quotient and a remainder.
• Analyze how factoring a polynomial can help determine its divisibility.
  • Factoring a polynomial can provide valuable insights into its divisibility. If a polynomial can be factored into the form $(x - a)(x - b)(x - c) \dots$, then it is divisible by each of the linear factors $(x - a)$, $(x - b)$, $(x - c)$, and so on. This is because, according to the Remainder Theorem, if a polynomial is divisible by a linear factor $(x - a)$, then the value of the polynomial when $x = a$ must be zero. By factoring the polynomial, you can identify the specific linear expressions that divide the polynomial evenly, which is crucial in the context of polynomial division.

Related terms

Factor: A factor is a number or expression that divides another number or expression without leaving a remainder.

Remainder Theorem:

The Remainder Theorem states that if a polynomial $P(x)$ is divided by $(x - a)$, then the remainder is $P(a)$.

Polynomial Division: Polynomial division is the process of dividing one polynomial by another, resulting in a quotient and a remainder.