Chapter 11.5, Problem 4AYU

To determine

True or False.

Answer to Problem 4AYU

Solution:

True.

Explanation of Solution

Given:

Every polynomial with real numbers as coefficients can be factored into products of linear and/or irreducible quadratic factors.

Every complex polynomial function of degree n has exactly n zeros and can be factored into a product of n linear factors. If its coefficient are real then those zeros that are complex numbers always occur as conjugate pairs. As a result, if $r = a + bi$ is a complex zero, so is $\overline{r} = a - bi$. Consequently, when the linear factors $x - r$ and $x - \overline{r}$ and of function are multiplied we have

$\begin{array}{lll}\left(x - r\right)\left(x - \overline{r}\right)\hfill & = \hfill & x^2 - \left(r + \overline{r}\right)x + r\overline{r}\hfill \\ \hfill & = \hfill & x^2 - 2ax + a^2 + b^2\hfill \end{array}$

This second degree polynomial has real coefficient and is irreducible (over real numbers). Thus it is true that the factors of function are either linear or irreducible quadratic factors.