Fundamental Theorem of Algebra

Learning Objectives State the Fundamental Theorem of Algebra and its corollaries. Determine the exact number of roots (including complex and repeated roots) for any polynomial equation. Apply the Complex Conjugate Root Theorem to find additional roots of a polynomial. Express a polynomial as a product of its linear factors over the set of complex numbers. Construct a polynomial function of least degree with given real and complex roots. Solve polynomial equations of degree three or higher by utilizing given complex or irrational roots. Ever wondered if every polynomial equation, no matter how complicated, has a solution? 🤔 The answer is yes, and this theorem proves it! The Fundamental Theorem of Algebra is a cornerstone concept that guarantees the existence of roots for an...

TermDefinitionExample Polynomial FunctionA function of the form P(x) = a_n x^n + a_{n-1}x^{n-1} + ... + a_1 x + a_0, where 'n' is a non-negative integer and the coefficients 'a' are real numbers.P(x) = 3x^4 - 5x^2 + 2x - 7 is a polynomial of degree 4. Degree of a PolynomialThe highest exponent of the variable in a polynomial.The polynomial P(x) = 2x^5 - x^3 + 8 has a degree of 5. Root (or Zero)A value 'c' for the variable 'x' such that P(c) = 0. Roots can be real or complex numbers.For P(x) = x^2 - 4, the roots are x = 2 and x = -2 because P(2) = 0 and P(-2) = 0. Complex NumberA number of the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, satisfying i^2 = -1.3 + 2i is a complex number, wh...

The Fundamental Theorem of Algebra Every non-constant single-variable polynomial P(x) with complex coefficients has at least one complex root. This is an existence theorem. It guarantees that a solution exists in the complex number system for any polynomial equation P(x) = 0, as long as the degree is 1 or higher. The N-th Degree Polynomial Corollary Every polynomial P(x) of degree n > 0 has exactly n roots in the complex number system, provided that roots are counted with their multiplicity. This is the most practical application of the theorem. It tells you exactly how many roots to look for. The degree of the polynomial equals the total number of roots (real and complex combined). Complex Conjugate Root Theorem If a polynomial P(x) has real coefficients, and if a...