Question

Linear and Quadratic Factors. A polynomial P is given (a)

Factors and multiples
Linear and Quadratic Factors. A polynomial P is given (a) Factor P into linear and irreducible quadratic factors with real coefficients. (b) Factor P completely into linear factors with complex coefficients.
$$\displaystyle{P}{\left({x}\right)}={x}^{{{4}}}+{8}{x}^{{{2}}}+{16}$$

2021-08-15
a) To find: The factor of the polynomial into linear and irreducible quadratic factor with real coefficients.
Consider, $$\displaystyle{P}{\left({x}\right)}={x}^{{{4}}}+{8}{x}^{{{2}}}+{16}$$
$$\displaystyle\Rightarrow{P}{\left({x}\right)}={x}^{{{4}}}+{4}{x}^{{{2}}}+{4}{x}^{{{2}}}+{16}$$
$$\displaystyle\Rightarrow{P}{\left({x}\right)}={x}^{{{2}}}{\left({x}^{{{2}}}+{4}\right)}+{4}{\left({x}^{{{2}}}+{4}\right)}$$
$$\displaystyle\Rightarrow{P}{\left({x}\right)}={\left({x}^{{{2}}}+{4}\right)}{\left({x}^{{{2}}}+{4}\right)}$$
Therefore, $$\displaystyle{P}{\left({x}\right)}={\left({x}^{{{2}}}+{4}\right)}{\left({x}^{{{2}}}+{4}\right)}$$.
b) To find: The factor completely into linear factors with complex coefficients.
Consider, $$\displaystyle{P}{\left({x}\right)}={x}^{{{4}}}+{8}{x}^{{{2}}}+{16}$$
$$\displaystyle\Rightarrow{P}{\left({x}\right)}={x}^{{{4}}}+{4}{x}^{{{2}}}+{4}{x}^{{{2}}}+{16}$$
$$\displaystyle\Rightarrow{P}{\left({x}\right)}={x}^{{{2}}}{\left({x}^{{{2}}}+{4}\right)}+{4}{\left({x}^{{{2}}}+{4}\right)}$$
$$\displaystyle\Rightarrow{P}{\left({x}\right)}={\left({x}^{{{2}}}+{4}\right)}{\left({x}^{{{2}}}+{4}\right)}$$
$$\displaystyle\Rightarrow{P}{\left({x}\right)}={\left({x}-{2}{i}\right)}{\left({x}-{2}{i}\right)}{\left({x}+{2}{i}\right)}{\left({x}-{2}{i}\right)}$$
Therefore, $$\displaystyle{P}{\left({x}\right)}={\left({x}-{2}{i}\right)}{\left({x}-{2}{i}\right)}{\left({x}+{2}{i}\right)}{\left({x}-{2}{i}\right)}$$.