Conjugate root theorems

Learning Objectives State the Complex Conjugate Root Theorem and the Irrational Conjugate Root Theorem. Identify the conjugate pair of a given complex or irrational root. Construct a polynomial of least degree with real coefficients from a given set of roots. Use the conjugate root theorems to find all roots of a polynomial when one complex or irrational root is known. Solve polynomial equations by applying the conjugate root theorems in conjunction with polynomial division. Explain why the theorems require polynomials to have real or rational coefficients. Ever solve for a polynomial's roots and find a strange one like 3 + 2i, then wonder if its 'twin' is hiding somewhere in the equation? 🤔 Let's find out! This tutorial explores the Conjugate Root Theo...

TermDefinitionExample Polynomial with Real CoefficientsA polynomial P(x) = a_n x^n + ... + a_1 x + a_0 where all the coefficients (a_n, ..., a_0) are real numbers.P(x) = 3x^4 - 5x^2 + 2 is a polynomial with real coefficients. P(x) = 2x^2 + ix - 1 is not. Root (or Zero)A value 'c' such that when you substitute it for the variable in a polynomial, the result is zero. That is, P(c) = 0.For P(x) = x^2 - 4, the roots are x = 2 and x = -2 because P(2) = 0 and P(-2) = 0. Complex ConjugateFor a complex number z = a + bi, its complex conjugate, denoted as z̄, is a - bi. The real part is the same, but the sign of the imaginary part is flipped.The complex conjugate of 5 + 3i is 5 - 3i. The conjugate of -7i is 7i. Irrational ConjugateFor a number of the form a + b√c where √c is an irrationa...

Complex Conjugate Root Theorem If P(x) is a polynomial with real coefficients, and z = a + bi is a root of P(x), then its complex conjugate z̄ = a - bi is also a root of P(x). Use this theorem when you are given a polynomial with real coefficients and find or are given one complex root. It automatically gives you a second root for free, simplifying the process of finding all roots. Irrational Conjugate Root Theorem If P(x) is a polynomial with rational coefficients, and a + b√c is a root of P(x) where √c is irrational, then its irrational conjugate a - b√c is also a root of P(x). Similar to the complex version, use this when you have a polynomial with rational coefficients and one irrational root. This theorem provides the second, conjugate root.