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Chapter 7: Problem 2

Give an example of each of the following. a. A simple linear factor b. A repeated linear factor c. A simple irreducible quadratic factor d. A repeated irreducible quadratic factor

Short Answer

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Question: Provide an example of each type of factor and explain its properties. Answer: (a) A simple linear factor is (x - 3). It has a single root at x = 3. (b) A repeated linear factor is (x - 2)^2. It has a root at x = 2 with multiplicity 2. (c) A simple irreducible quadratic factor is x^2 + 1. It has no real roots, as the discriminant is -3 < 0. (d) A repeated irreducible quadratic factor is (x^2 + 1)^2. It has no real roots, and it has a multiplicity of 2.

Step by step solution

01

(a) Simple Linear Factor

A simple linear factor is a linear expression with a single root. It has the general form \((x - a)\), where \(a\) is a constant. An example of a simple linear factor is \((x - 3)\). This factor has a single root at \(x = 3\).
02

(b) Repeated Linear Factor

A repeated linear factor is a linear expression raised to a certain power greater than 1. It has the general form \((x - a)^n\), where \(a\) is a constant and \(n\) is an integer greater than 1. An example of a repeated linear factor is \((x - 2)^2\). This factor has a root at \(x = 2\) with multiplicity 2.
03

(c) Simple Irreducible Quadratic Factor

A simple irreducible quadratic factor is a quadratic expression that cannot be factored further and has no real roots. It has the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and the discriminant \(b^2 - 4ac < 0\). An example of a simple irreducible quadratic factor is \(x^2 + 1\). This factor has no real roots, as the discriminant is \((-1)^2 - 4(1)(1) = -3 < 0\).
04

(d) Repeated Irreducible Quadratic Factor

A repeated irreducible quadratic factor is an irreducible quadratic expression raised to a certain power greater than 1. It has the general form \((ax^2 + bx + c)^n\), where \(a\), \(b\), and \(c\) are constants, the discriminant \(b^2 - 4ac < 0\), and \(n\) is an integer greater than 1. An example of a repeated irreducible quadratic factor is \((x^2 + 1)^2\). This factor has no real roots, as the discriminant is \((-1)^2 - 4(1)(1) = -3 < 0\), and it has a multiplicity of 2.

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