Problem 2 - Calculus Early Transcendentals

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

Step-by-Step Solution

Answer

Question: Provide an example for each of the following types of factors: a. Simple linear factor b. Repeated linear factor c. Simple irreducible quadratic factor d. Repeated irreducible quadratic factor Answer: a. (2x + 3) b. (x - 4)^2 c. (x^2 + 2x + 2) d. (x^2 + x + 1)^2

Step 1: a. Simple linear factor

A simple linear factor is a linear expression with a single power and can be represented as \((ax + b)\), where \(a \neq 0\). An example of a simple linear factor is: \((2x + 3)\).

Step 2: b. Repeated linear factor

A repeated linear factor is a linear expression raised to a power greater than 1 and can be represented as \((ax + b)^n\), where \(a \neq 0\) and \(n > 1\). An example of a repeated linear factor is: \((x - 4)^2\).

Step 3: c. Simple irreducible quadratic factor

A simple irreducible quadratic factor is a quadratic expression that cannot be factored any further and can be represented as \((ax^2 + bx + c)\), where \(a \neq 0\) and the discriminant \(b^2 - 4ac < 0\). An example of a simple irreducible quadratic factor is: \((x^2 + 2x + 2)\).

Step 4: d. Repeated irreducible quadratic factor

A repeated irreducible quadratic factor is an irreducible quadratic expression raised to a power greater than 1 and can be represented as \((ax^2 + bx + c)^n\), where \(a \neq 0\), the discriminant \(b^2 - 4ac < 0\), and \(n > 1\). An example of a repeated irreducible quadratic factor is: \((x^2 + x + 1)^2\).