Flashcard +, -, X Complex Numbers & Powers of i

Front

What are complex numbers?

Back

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit, defined as the square root of -1.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as i = √(-1). It is used to represent the square root of negative numbers.

Front

How do you add complex numbers?

Back

To add complex numbers, combine their real parts and their imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.

Front

How do you multiply complex numbers?

Back

To multiply complex numbers, use the distributive property (FOIL method). For example, (a + bi)(c + di) = ac + adi + bci + bdi², and since i² = -1, simplify accordingly.

Front

What is the conjugate of a complex number?

Back

The conjugate of a complex number a + bi is a - bi. It is used to simplify division of complex numbers.

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How do you find the zeros of a polynomial function?

Back

To find the zeros of a polynomial function, set the function equal to zero and solve for the variable. This may involve factoring, using the quadratic formula, or other methods.

Front

What is the quadratic formula?

Back

The quadratic formula is used to find the solutions (zeros) of a quadratic equation ax² + bx + c = 0. It is given by x = (-b ± √(b² - 4ac)) / (2a).

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What does it mean to factor a polynomial completely?

Back

Factoring a polynomial completely means expressing it as a product of its irreducible factors, which cannot be factored further over the given number system.

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What is the difference between a polynomial and a complex polynomial?

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A polynomial is an expression consisting of variables and coefficients, while a complex polynomial includes complex numbers as coefficients or roots.

Front

How do you simplify the expression (5 + 2i)(-4 + 3i)?

Back

To simplify (5 + 2i)(-4 + 3i), use the distributive property: = 5*(-4) + 5*(3i) + 2i*(-4) + 2i*(3i) = -20 + 15i - 8i + 6 = -26 + 7i.