Polynomial Long Division

Definition

Polynomial long division is a method used to divide one polynomial by another polynomial. It is a systematic process of dividing the dividend polynomial by the divisor polynomial, similar to the long division algorithm used for dividing integers. This technique is particularly useful for finding the zeros or roots of a polynomial function.

5 Must Know Facts For Your Next Test

1. Polynomial long division is a technique used to find the quotient and remainder when dividing one polynomial by another.
2. The process of polynomial long division involves repeatedly dividing the leading term of the dividend by the leading term of the divisor, and then subtracting the resulting multiple of the divisor from the dividend.
3. The zeros of a polynomial function are the values of the variable for which the function is equal to zero, and they can be found using polynomial long division.
4. The Remainder Theorem states that when a polynomial $P(x)$ is divided by $(x - a)$, the remainder is equal to $P(a)$, which can be used to find the zeros of a polynomial function.
5. Polynomial long division is a useful tool for factoring polynomials and solving polynomial equations, which are important concepts in the study of polynomial functions.

Review Questions

• Explain the step-by-step process of performing polynomial long division.
  • The process of polynomial long division involves the following steps: 1) Divide the leading term of the dividend by the leading term of the divisor to obtain the first term of the quotient. 2) Multiply the divisor by the first term of the quotient and subtract the result from the dividend. 3) Bring down the next term of the dividend and repeat steps 1 and 2 until the degree of the remainder is less than the degree of the divisor. The final result is the quotient and the remainder.
• Describe 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 the linear factor $(x - a)$, where $a$ is a potential zero of the function. If the remainder is zero, then $a$ is a zero of the polynomial function. If the remainder is non-zero, then $a$ is not a zero of the polynomial function. This process can be repeated for different values of $a$ to find all the zeros of the polynomial function.
• Explain the connection between polynomial long division and the Remainder Theorem, and how this relationship can be used to solve polynomial equations.
  • The Remainder Theorem states that when a polynomial $P(x)$ is divided by $(x - a)$, the remainder is equal to $P(a)$. This means that if $P(a) = 0$, then $(x - a)$ is a factor of $P(x)$, and $a$ is a zero of the polynomial function. By using polynomial long division to divide a polynomial by $(x - a)$ and finding the remainder, you can determine if $a$ is a zero of the polynomial function. This process can be repeated for different values of $a$ to find all the zeros of the polynomial function, which is an essential step in solving polynomial equations.

Related terms

Polynomial Function:

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

Zeros of a Polynomial Function: The zeros of a polynomial function are the values of the variable for which the function is equal to zero. These zeros represent the points where the graph of the function intersects the x-axis.

Remainder Theorem:

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