Question

How do you add subtract multiply and divide complex numbers and polynomials

Answer 1

To operate on complex numbers, add or subtract real and imaginary parts separately, multiply using the distributive property, and divide by multiplying by the conjugate of the denominator. Polynomial operations similarly involve combining like terms, using the distributive property, and performing long or synthetic division.

Step-by-step explanation:

Adding, Subtracting, Multiplying, and Dividing Complex Numbers and Polynomials

To add complex numbers, combine the real parts and the imaginary parts separately. For example, (3 + 2i) + (1 + 4i) = (3 + 1) + (2i + 4i) = 4 + 6i. For subtracting complex numbers, similarly subtract the real and imaginary parts: (3 + 2i) - (1 + 4i) = (3 - 1) + (2i - 4i) = 2 - 2i.

To multiply complex numbers, use the distributive property (FOIL), and remember that i² = -1. For instance, (3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i² = 3 + 14i - 8 = -5 + 14i.

For dividing complex numbers, multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. For example, (3 + 2i) ÷ (1 + 4i), multiply numerator and denominator by (1 - 4i) to get [(3 + 2i)(1 - 4i)] ÷ [(1 + 4i)(1 - 4i)] = (3 - 12i + 2i - 8i²) ÷ (1 - 16i²) = (11 - 10i) ÷ 17 = (11÷ 17) - (10i÷ 17).

Polynomial operations are similar, but instead of imaginary units, we work with variables. For adding polynomials, combine like terms. For subtracting polynomials, change the sign of each term in the polynomial being subtracted and combine like terms. Multiplying polynomials also requires the distributive property, and for division, you can use either long division or synthetic division depending on the polynomials involved.

Answer 2

A complex number is the sum (or difference) of two parts: real and imaginary.To add or subtract two complex numbers, need to add\subtract separately the real part and imaginary separately. In the end, you will receive the imaginary and real part of the result.

More difficult to divide complex numbers. 1. We must find the complex conjugate of the denominator 2. Then, we must multiply the numerator and denominator by that conjugate and simplify the expression.The complex conjugate have the part equal to given, and the imaginary part is opposite to given.

To multiply complex numbers, we must multiply it like the two expressions in brackets, and simplify the result.

Addition subtraction and multiplication of polynomials is also made as expressions in parentheses between them. During the operation it is necessary to cause such members.To divide the polynomial by the polynomial you need to record them so as if you divide the numbers in a column, and perform the division by the same rules.