Fundamental Theorem of Algebra

Learning Objectives State the Fundamental Theorem of Algebra and its corollary. Determine the total number of roots for a polynomial by identifying its degree. Apply the Complex Conjugate Root Theorem to identify pairs of complex roots. Find all real and complex roots of a polynomial function. Construct a polynomial function in standard form given its roots. Understand the concept of multiplicity of roots. Ever wondered if every polynomial equation has a solution, even if you can't see it on a graph? 🤔 This theorem guarantees it! The Fundamental Theorem of Algebra is a cornerstone concept that guarantees every non-constant polynomial has at least one root in the complex number system. This lesson will show you how to find every single root, both real and complex, for...

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 Degree of a PolynomialThe highest exponent of the variable in a polynomial.The polynomial P(x) = 3x^4 - 5x^2 + 2x - 7 has a degree of 4. Root (or Zero)A value 'c' for the variable 'x' that makes the polynomial equal to zero, i.e., P(c) = 0. On a graph, real roots are the x-intercepts.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 written in the form a + bi, where 'a' is the real part, 'b' is the imaginary part, and i is the imaginary unit (i^2 = -1).3...

The Fundamental Theorem of Algebra Every non-constant single-variable polynomial with complex coefficients has at least one complex root. This is the core guarantee. It tells us that a solution always exists, as long as we are working within the complex number system. You don't solve with this theorem, but it's the foundation for why we can always find roots. The n-Roots Corollary Every polynomial of degree n > 0 has exactly n roots, counting multiplicity, in the set of complex numbers. This is the practical application of the theorem. It tells you exactly how many roots to look for. A polynomial of degree 3 has 3 roots, a polynomial of degree 5 has 5 roots, and so on. The Complex Conjugate Root Theorem If a polynomial P(x) has real coefficients, and if a...