1. Using the Horner s scheme, find the incomplete quotient and remainder of dividing the polynomial A(x

Horner's scheme and polynomial division

Инструкция:
Horner's scheme is a method used to efficiently evaluate polynomials and perform polynomial division. It allows us to find the incomplete quotient and remainder when dividing a polynomial by a binomial.

To use Horner's scheme, we start by setting up a synthetic division table. The coefficients of the polynomial A(x) are placed in the first row, and the constant term of the binomial B(x) is placed on the left side of the table. We then perform division and multiplication operations to fill in the remaining cells of the table.

For example, in the first task, we have A(x) = x^3 + 3x^2 + 3x + 1 and B(x) = x + 1. We set up the synthetic division table as follows:

| 1 | 3 | 3 | 1
-1 | | | |
____|_____|____|____|

Next, we bring down the first coefficient, which is 1. Then, we multiply the constant term of B(x), which is -1, by this coefficient and write the result in the next cell. In this case, -1 * 1 = -1.

| 1 | 3 | 3 | 1
-1 | -1 | | |
____|_____|___|____|

We add the corresponding coefficients in the first row and the result is written below the line.

| 1 | 3 | 3 | 1
-1 | -1 | 2 | |
____|_____|___|____|

We continue this process until we reach the end of the row.

| 1 | 3 | 3 | 1
-1 | -1 | 2 | -1 | -2 |
____|_____|___|____|

The numbers in the last row represent the coefficients of the incomplete quotient: 1, 2, -1. The remainder is the last number in the second-to-last row, -2.

Дополнительный материал:
1. A(x) = x^3 + 3x^2 + 3x + 1; B(x) = x + 1.
Incomplete quotient: 1, 2, -1. Remainder: -2.

Совет:
To understand and apply Horner's scheme effectively, it's important to grasp the concept of polynomial division and be familiar with basic operations like multiplication and addition/subtraction. Practice using different examples to reinforce your understanding. Also, pay attention to the signs when performing multiplication and addition/subtraction in the scheme.

Упражнение:
1. Divide the polynomial A(x) by the binomial B(x):
A(x) = 2x^3 - 19x^2 + 32x + 21; B(x) = x - 7.