How would you multiply this equation b) $2(x - i)(x + i)(x + 5)$This expression represents a polynomial expanded from multiple binomials. It involves complex numbers, as indicated by the presence of $i$, which is the imaginary unit ($i = \sqrt{-1}$). The expression features the product of three binomials and an initial coefficient of 2. Understanding this expression is crucial for solving complex roots and polynomial equations. For educational purposes, it illustrates how complex conjugates ($x - i$ and $x + i$) are used in algebraic operations.

Answered: ) 2(х — i) (х + i) (x + 5)

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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How would you multiply this equation

b) $2(x - i)(x + i)(x + 5)$

This expression represents a polynomial expanded from multiple binomials. It involves complex numbers, as indicated by the presence of $i$, which is the imaginary unit ($i = \sqrt{-1}$). The expression features the product of three binomials and an initial coefficient of 2. Understanding this expression is crucial for solving complex roots and polynomial equations. For educational purposes, it illustrates how complex conjugates ($x - i$ and $x + i$) are used in algebraic operations.

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We have to multiply $2 (x - i) (x + i) (x + 5)$

$\begin{array}{rcl} 2 (x - i) (x + i) (x + 5) & = & 2 (x - i) [x (x + 5) + i (x + 5)] \\ = & 2 (x - i) [x^{2} + 5 x + x i + 5 i] \\ = & 2 (x - i) [x^{2} + 5 x + (x + 5) i] \\ = & 2 [x (x^{2} + 5 x + (x + 5) i) - i (x^{2} + 5 x + (x + 5) i)] \\ = & 2 [x^{3} + 5 x^{2} + x^{2} i + 5 x i - x^{2} i - 5 x i + (x + 5) i^{2}] \\ = & 2 [x^{3} + 5 x^{2} + (x + 5) \cdot - 1] \\ = & 2 [x^{3} + 5 x^{2} - x - 5] \\ = & 2 x^{3} + 10 x^{2} - 2 x - 10 \end{array}$

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