Chapter 2.5, Problem 2E

To determine

A quadratic factor that cannot be factored as a product of linear factors containing real numbers is said to be _____ over the _____.

Answer to Problem 2E

Irreducible, real

Explanation of Solution

A quadratic factor that cannot be factored as a product of linear factors containing real numbers is said to be irreducible over the real.

For example, consider a function $f(x)=x^4+25x^2+144$ .

Calculation:

  $\begin{array}{l}\Rightarrow f(x)=x^4+25x^2+144\\ \Rightarrow f(x)=x^4+9x^2+16x^2+144\\ \Rightarrow f(x)=x^2(x^2+9)+16(x^2+9)\\ \Rightarrow f(x)=(x^2+9)(x^2+16)\end{array}$

Here, the quadratic factors $x^2+9\text{and}{x}^2+16$ cannot be factored further as a product of linear factors, though they contain real numbers.

So, they are said to be irreducible over the real.