This post walks through how to solve a fourth-degree polynomial equation (quartic equation) and identify all real and imaginary solutions. It includes a key reminder for students not to confuse cube root shortcuts with proper factorization and introduces correct techniques using the difference of squares and factoring.

Example solved in the image:
x⁴ − 256 = 0

Key steps covered:

• Recognizing that 256 = 4⁴

• Applying the identity for difference of squares: x² − a² = (x − a)(x + a)

• Continuing the factorization down to quadratics: (x² − 16)(x² + 16)

• Solving the real roots: x = ±4

• Solving the imaginary roots: x = ±4i

Final solution set:
x = {–4, –4i, 4i, 4}

This visual is taken directly from the Ultimate Crash Course for STEM Majors, which prepares students for algebra, calculus, and complex number theory.

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