Question

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

Question: Provide an example for each type of factor: a) simple linear factor, b) repeated linear factor, c) simple irreducible quadratic factor, and d) repeated irreducible quadratic factor. Answer: a) (x-2), b) (x-3)^2, c) (x^2 + 1), d) (x^2 + 1)^2

Step-by-step solution

A simple linear factor

A simple linear factor is a linear term which is not repeated in a polynomial. For example, (x-2) is a simple linear factor, because it is a linear term that occurs only once.

A repeated linear factor

A repeated linear factor is a linear term that occurs more than once in a polynomial. For example, (x-3)^2 is a repeated linear factor because the term (x-3) is repeated twice.

A simple irreducible quadratic factor

A simple irreducible quadratic factor is a quadratic term that cannot be factored further and occurs only once in a polynomial. For example, (x^2 + 1) is a simple irreducible quadratic factor, since it cannot be factored further (it has no real roots) and occurs only once.

A repeated irreducible quadratic factor

A repeated irreducible quadratic factor is a quadratic term that cannot be factored further and occurs more than once in a polynomial. For example, (x^2 + 1)^2 is a repeated irreducible quadratic factor, because it cannot be factored further (no real roots) and the term (x^2 + 1) is repeated twice.

Key concepts

Simple Linear Factor

When we delve into the world of algebra, understanding factors is crucial. A simple linear factor is one of the building blocks of polynomial expressions. Picture a simple linear factor as a puzzle piece that, when joined with others, forms a complete mathematical picture or polynomial. Specifically, a simple linear factor refers to a linear expression of the form \(x - a\), where \(a\) is a constant. The crucial aspect here is its uniqueness within the polynomial—it occurs exactly once.

To illustrate, if we take a polynomial like \(x^2 - 2x + 1\), we can factor it to \(x - 1)(x - 1)\). However, the simple linear factor here is just \(x - 1\), representing a unique piece that makes up the full expression. It's like identifying a single distinct ingredient in a recipe—a necessary component that you won't find repeated.

Repeated Linear Factor

In contrast to the unique nature of a simple linear factor, a repeated linear factor is one that makes several appearances in a polynomial, like a recurring character in a story. It takes the form of \(x - b)^n\), where \(b\) is a constant and \(n\) represents the number of times the factor is repeated—a power greater than 1. Think of this as a chorus in a song, where the same lines are repeated to emphasize their importance.

An example of this would be \(x^4 - 4x^3 + 6x^2 - 4x + 1\), which can be broken down into \(x - 1)^4\), showing that the factor \(x - 1\) is indeed repeated four times. This repetition has implications when solving algebraic equations, as it hints at multiple, identical roots.

Simple Irreducible Quadratic Factor

Moving up in complexity, we find the simple irreducible quadratic factor. This is a quadratic expression that cannot be factored into real-number linear factors and appears only once in a polynomial. The term 'irreducible' signals that you can't break it down further using real numbers. Typically this factor has the form \(x^2 + bx + c\), where the discriminant \((b^2 - 4ac)\) is less than zero, leading to complex roots.

For example, \(x^2 + 2x + 5\) is an irreducible quadratic factor of the polynomial \(x^4 + 4x^3 + 14x^2 + 12x + 5\). The quadratic itself cannot be simplified or 'reduced' using basic factoring techniques which makes it a fundamental part of the polynomials structure. It's like a compound in chemistry that doesn’t break down into something simpler under normal conditions.

Repeated Irreducible Quadratic Factor

Imagine not just one, but several layers of complexity added to our polynomial—this is where the repeated irreducible quadratic factor comes into play. A repeated irreducible quadratic factor maintains its irreducible nature, but unlike its simple counterpart, it shows up more than once. It is of the form \( (x^2 + bx + c)^m \), where \(m\) is an integer greater than 1, indicating the factor’s repetition.

An exemplar of such a factor could be \( (x^2 + x + 1)^2 \), which repeats the complex quadratic \(x^2 + x + 1\) twice. The presence of this factor in a polynomial suggests a layered complexity, almost like having a repeated refrain in a complex piece of music, with the notes intricately woven together to create a harmonious tune.