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Find an explicit complex number a + bi which is a root of x³ – 1 but is not a root of x4 – 1. (Hint: find a solution to (a + bi)² = i). Use the quadratic formula to find the complex roots (zeroes) of the polynomials x² + 5x + 100 and (1 + i)x² + 2x + 1 — i, expressing your answers in the form a + bi where a, b € R. Factor each polynomial as a product of linear (degree one) polynomials.

Find an explicit complex number a + bi which is a root of x³ – 1 but is not a root of x4 – 1. (Hint: find a solution to (a + bi)² = i). Use the quadratic formula to find the complex roots (zeroes) of the polynomials x² + 5x + 100 and (1 + i)x² + 2x + 1 — i, expressing your answers in the form a + bi where a, b € R. Factor each polynomial as a product of linear (degree one) polynomials.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Find an explicit complex number \(a + bi\) which is a root of \(x^8 - 1\) but is not a root of \(x^4 - 1\). (Hint: find a solution to \((a + bi)^2 = i\)). Use the quadratic formula to find the complex roots (zeroes) of the polynomials \(x^2 + 5x + 100\) and \((1 + i)x^2 + 2x + 1 - i\), expressing your answers in the form \(a + bi\) where \(a, b \in \mathbb{R}\). Factor each polynomial as a product of linear (degree one) polynomials.
Transcribed Image Text:**Problem Statement:** Find an explicit complex number \(a + bi\) which is a root of \(x^8 - 1\) but is not a root of \(x^4 - 1\). (Hint: find a solution to \((a + bi)^2 = i\)). Use the quadratic formula to find the complex roots (zeroes) of the polynomials \(x^2 + 5x + 100\) and \((1 + i)x^2 + 2x + 1 - i\), expressing your answers in the form \(a + bi\) where \(a, b \in \mathbb{R}\). Factor each polynomial as a product of linear (degree one) polynomials.
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