Complex Numbers & Polynomials- Both Basic & Advanced

Define complex numbers as a+ib and explore their real and imaginary parts, operations (addition, subtraction, multiplication), and equality, with a focus on complex conjugates and the Argand diagram.

Learn how to perform addition, subtraction, multiplication, and division of complex numbers, express results as a + i b, and apply i^2 = -1 to simplify.

Solve quadratics with real and complex roots by factoring x^2-4 as a difference of squares. Use the quadratic formula; a negative discriminant yields 5±2i.

Explore complex conjugates: real parts remain unchanged while the imaginary parts negate, making conjugate pairs mirror images across the real axis on the Argand diagram.

Explore how negative discriminants produce complex conjugate roots a±bi and how their sum and product yield the quadratic x^2 − (sum) x + (product) = 0.

Outline five key properties of complex conjugates: double conjugation returns the original number, conjugation distributes over addition and multiplication, powers respect conjugation, and z times conj(z) is real.

Learn how to multiply polynomials using term-by-term expansion and collect like terms, then divide polynomials by another polynomial, identifying dividend, divisor, quotient, and remainder.

Explore real polynomials by defining polynomials and polynomial functions, performing addition, subtraction, scalar multiplication, and multiplication and division, and understand degree and linear, quadratic, and cubic cases.

Learn to add and subtract polynomials by combining like terms, applying correct signs, and perform scalar multiplication with examples using x^2, x, and constants.

Explore zeros and roots of polynomials and how factors reveal them. Practice factoring and solving product equations to find x-values, such as x+2 being a factor when x^2+4x+4=0.

Apply polynomial equality to two quadratics by expanding and equating corresponding coefficients. Solve to find A = -B and B^2 = 6, giving (A,B) = (√6, −√6) or (−√6, √6).

The remainder theorem gives f(k) as remainder when f(x) is divided by x minus k, and factor theorem says if E is a zero, x minus E is a factor.

Explains how the sum and product of roots relate to polynomial coefficients, from quadratic to general degree, and shows a seven-degree example with sum zero and product nine.

Explore quadratic graphs by comparing standard form and factored form y=a(x-alpha)(x-beta), with alpha and beta as x-intercepts. See how a governs concavity, vertex, and width, and fixes a in y=(2/7)(x-4)(x+1).

Explore cubic functions by analyzing x intercepts alpha, beta, gamma and representing as (x−alpha)(x−beta)(x−gamma) with coefficients A, B, or C. Learn how extra points determine constants and explain repeated roots.

Explore quartic functions and their four roots, represented by (x - alpha)(x - beta)(x - gamma)(x - delta), including repeated roots and the fundamental theorem of algebra.

Learn to find the argument of a complex number from its real and imaginary parts via arctan, adjusting for quadrant to measure angle with the positive x-axis in 0–360° (0–2π).

Express a complex number in polar form using modulus and argument; for z = 2 + 3i, modulus is sqrt(13) and the argument is arctan(3/2).

Convert cartesian complex numbers to polar form by computing modulus and argument. Express as modulus cis theta, e.g., 2i = 2 cis pi/2 and 1-8i = sqrt(2) cis 7pi/4.

Explore polar form properties of complex numbers, including modulus invariance under conjugation, multiplication and division rules using cis notation, and the role of angles in complex multiplication.

Compute modulus and argument from coordinates, express a complex number as r cis theta, and multiply by another using modulus multiplication and angle addition, with angle reduction and visual interpretation.

Compute the modulus and argument of z = cos φ + i sin φ − 1. The modulus is 2 sin(φ/2) and the argument π/2 + φ/2.