What does it mean to solve over a set of complex numbers?
Over the complex numbers, every polynomial (with real-valued coefficients) can be factored into a product of linear factors. We can state this also in root language: Over the complex numbers, every polynomial of degree n (with real-valued coefficients) has n roots, counted according to their multiplicity.
What is the set of complex numbers?
A complex number is a number that can be written in the form a + b i a + bi a+bi, where a and b are real numbers and i is the imaginary unit defined by i 2 = − 1 i^2 = -1 i2=−1. The set of complex numbers, denoted by C, includes the set of real numbers (R) and the set of pure imaginary numbers.
How do you find the real solution?
If the value of the discriminant is positive, there are two real solutions for x, meaning the graph of the solution has two distinct x-intercepts. If the value of the discriminant is zero, there is one real solution for x, meaning the graph of the solution has one x-intercept.
Are complex solutions real solutions?
The expression b2 − 4ac is called the discriminant, and can be used to determine whether the solutions are real, repeated, or complex: 1) If the discriminant is less than zero, the equation has two complex solution(s). 2) If the discriminant is equal to zero, the equation has one repeated real solution(s).
What is 3i equal to?
Remember that a complex number has the form a + bi. You need to figure out what a and b need to be. Since −3i is an imaginary number, it is the imaginary part (bi) of the complex number a + bi. This imaginary number has no real parts, so the value of a is 0….
| Imaginary Numbers | |
|---|---|
| 3i (b = 3) | −672i (b = −672) |
| (b = ) | (b = ) |
What are real and complex solutions?
If the discriminant equals 0, then the equation has one real solution, a double root. If the discriminant is less than 0, then the equation has two complex solutions. If the discriminant is greater than 0, then the equation has two real solutions.