songoku said:Homework Statement: a. Solve w^3 = 1 in de moivre form
b. You must use result from (a) to solve (z + i)^3 + (z - i)^3 = 0
Homework Equations: de moivre
I can do question (a). For question (b), I can not see the relation to question (a). Can we really do question (b) using result from (a)? Please give me little hint to relate them
Thanks
Math_QED said:Divide both sides in (b) by ##(z+i)^3## and use that ##-1=(-1)^3## to write the equation in the form
$$\left(\frac{i-z}{i+z}\right)^3=1$$
Then you can use (a) to proceed.
A polynomial is an algebraic expression that consists of variables and coefficients, combined using the operations of addition, subtraction, and multiplication. It can also include exponents, but the exponents must be positive integers.
Complex numbers are numbers that include both a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part with i being the imaginary unit (√-1).
Complex numbers allow us to find solutions to polynomials that cannot be solved using only real numbers. This is because some polynomials have roots that are complex numbers, and without using complex numbers, we would not be able to find these solutions.
To solve polynomials using complex numbers, we use the fundamental theorem of algebra, which states that every polynomial of degree n has n complex roots. We can then use techniques such as factoring, the quadratic formula, or synthetic division to find these roots.
Solving polynomials using complex numbers has many practical applications, including in engineering, physics, and computer science. It is used to model and solve problems in areas such as electrical circuits, signal processing, and fluid dynamics.