Polynomial Long Division

Definition

Polynomial long division is a method used to divide one polynomial by another polynomial. It involves a step-by-step process of dividing the terms of the dividend by the terms of the divisor, similar to the long division algorithm used for dividing integers.

5 Must Know Facts For Your Next Test

1. Polynomial long division is used to find the quotient and remainder when dividing one polynomial by another.
2. The process of polynomial long division involves dividing the leading term of the dividend by the leading term of the divisor, and then subtracting the result multiplied by the divisor from the dividend.
3. Polynomial long division is an important technique for solving polynomial equations, finding the zeros of polynomial functions, and working with partial fractions.
4. The remainder theorem states that the remainder when a polynomial $P(x)$ is divided by $(x - a)$ is $P(a)$.
5. The factor theorem states that if $P(a) = 0$, then $(x - a)$ is a factor of $P(x)$.

Review Questions

• Explain how polynomial long division can be used to find the zeros of a polynomial function.
  • Polynomial long division can be used to find the zeros of a polynomial function by dividing the polynomial by $(x - a)$, where $a$ is a potential zero of the function. If the remainder is zero, then $a$ is a zero of the polynomial function. This is a direct application of the remainder theorem. By repeatedly dividing the polynomial by $(x - a)$ for different values of $a$, you can determine all the zeros of the polynomial function.
• Describe how polynomial long division is used in the context of partial fractions.
  • Polynomial long division is a crucial step in the process of decomposing a rational function into partial fractions. When a rational function is in the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, polynomial long division is used to divide $P(x)$ by $Q(x)$. The resulting quotient and remainder are then used to determine the partial fraction decomposition of the original rational function, which is an important technique in calculus and other advanced mathematics.
• Analyze how the properties of polynomial long division, such as the remainder theorem and the factor theorem, can be used to solve polynomial equations and understand the behavior of polynomial functions.
  • The properties of polynomial long division, specifically the remainder theorem and the factor theorem, provide powerful tools for solving polynomial equations and analyzing the behavior of polynomial functions. The remainder theorem states that the remainder when a polynomial $P(x)$ is divided by $(x - a)$ is $P(a)$. This means that if $P(a) = 0$, then $(x - a)$ is a factor of $P(x)$, as stated by the factor theorem. By using polynomial long division to divide a polynomial by $(x - a)$ for various values of $a$, you can determine the roots or zeros of the polynomial function, which are essential for understanding its behavior and solving related equations.

Related terms

Polynomial:

A polynomial is an expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers.

Quotient:

The result of dividing one number by another, or in the case of polynomial long division, the result of dividing one polynomial by another.

Remainder:

The amount left over after a division operation, which in the case of polynomial long division, is the polynomial that is left after the division process is complete.