Question

Задание Даны многочлены: M= 2x1 + x³y - 3x²y² + 4xy³ - y N = -3x + 2x3y +5x²y2+y4 K = x+ - x³y - 2x²у2 + 4xy³ - 2y Найдите: M+N+к M-N+K M-N-K -М+N+К Запишите ответ в стандартном виде.​

Answer 1

Problem Analysis

We are given three polynomials: M, N, and K. We need to perform various operations with these polynomials and express the results in standard form.

Polynomial Definitions

The given polynomials are: - M = 2x + x³y - 3x²y² + 4xy³ - y - N = -3x + 2x³y + 5x²y² + y⁴ - K = x - x³y - 2x²y² + 4xy³ - 2y

Addition of Polynomials (M + N)

To find the sum of polynomials M and N, we add the corresponding terms together. The sum of M and N is: - M + N = (2x + x³y - 3x²y² + 4xy³ - y) + (-3x + 2x³y + 5x²y² + y⁴)

Subtraction of Polynomials (M - N)

To find the difference between polynomials M and N, we subtract the corresponding terms. The difference of M and N is: - M - N = (2x + x³y - 3x²y² + 4xy³ - y) - (-3x + 2x³y + 5x²y² + y⁴)

Subtraction of Polynomials (M - K)

To find the difference between polynomials M and K, we subtract the corresponding terms. The difference of M and K is: - M - K = (2x + x³y - 3x²y² + 4xy³ - y) - (x - x³y - 2x²y² + 4xy³ - 2y)

Subtraction of Polynomials (N - K)

To find the difference between polynomials N and K, we subtract the corresponding terms. The difference of N and K is: - N - K = (-3x + 2x³y + 5x²y² + y⁴) - (x - x³y - 2x²y² + 4xy³ - 2y)

Subtraction of Polynomials (M - N - K)

To find the difference between polynomials M, N, and K, we subtract the corresponding terms. The difference of M, N, and K is: - M - N - K = (2x + x³y - 3x²y² + 4xy³ - y) - (-3x + 2x³y + 5x²y² + y⁴) - (x - x³y - 2x²y² + 4xy³ - 2y)

Addition of Polynomials (M + N + K)

To find the sum of polynomials M, N, and K, we add the corresponding terms together. The sum of M, N, and K is: - M + N + K = (2x + x³y - 3x²y² + 4xy³ - y) + (-3x + 2x³y + 5x²y² + y⁴) + (x - x³y - 2x²y² + 4xy³ - 2y)

Standard Form

To express the results in standard form, we simplify the polynomials by combining like terms and arranging them in descending order of exponents.

The final expressions in standard form are: - M + N = 3x³y + 2x²y² + 8xy³ + y⁴ - y - M - N = 5x + 8x³y - 8x²y² + 3xy³ - y⁴ - M - K = x + 2x³y - x²y² + 0xy³ + y - N - K = -2x + 3x³y + 7x²y² + y⁴ + 2y - M - N - K = 5x + 8x³y - 8x²y² + 3xy³ - y⁴ - M + N + K = 3x³y + 2x²y² + 8xy³ + y⁴ - y

Note: The terms with a coefficient of 0 have been omitted from the final expressions.

Therefore, the answer in standard form is: - M + N = 3x³y + 2x²y² + 8xy³ + y⁴ - y - M - N = 5x + 8x³y - 8x²y² + 3xy³ - y⁴ - M - K = x + 2x³y - x²y² + y - N - K = -2x + 3x³y + 7x²y² + y⁴ + 2y - M - N - K = 5x + 8x³y - 8x²y² + 3xy³ - y⁴ - M + N + K = 3x³y + 2x²y² + 8xy³ + y⁴ - y