Presentation on theme: "Complex Numbers The imaginary number i is defined as so that Complex numbers are in the form a + bi where a is called the real part and bi is the imaginary."— Presentation transcript:
Complex Numbers The imaginary number i is defined as so that Complex numbers are in the form a + bi where a is called the real part and bi is the imaginary part.
Complex Numbers If a + bi is a complex number, its complex conjugate is a – bi. To add or subtract complex numbers, add or subtract the real parts and add or subtract the imaginary parts. To multiply two complex numbers, use FOIL, taking advantage of the fact that to simplify. To divide two complex numbers, multiply top and bottom by the complex conjugate of the bottom.
Complex Numbers Complex solutions to the Quadratic Formula When using the Quadratic Formula to solve a quadratic equation, you may obtain a result like, which you should rewrite as In general if a is positive.
Polynomial Roots (zeros) If f(x) is a polynomial of degree n, then f has precisely n linear factors: where c 1, c 2, c 3,… c n are complex numbers. This means that c 1, c 2, c 3,… c n are all roots of f(x), so that f(c 1 ) = f(c 2 ) = f(c 3 ) = … =f(c n ) = 0 Note: some of these roots may be repeated.
Polynomial Roots (zeros) For polynomial equations with real coefficients, any complex roots will occur in conjugate pairs. (If a + bi is a root, then a - bi is also a root)