Consider the following polynomials:

\(\displaystyle{3}{x}^{{4}}-{48}\)

To factorize the polynomials:

\(\displaystyle{3}{x}^{{4}}-{48}={3}{\left({x}^{{4}}-{16}\right)}\)

\(\displaystyle={3}{\left({\left({x}^{{2}}\right)}^{{2}}-{\left({4}\right)}^{{2}}\right)}\)

\(\displaystyle={3}{\left({x}^{{2}}+{4}\right)}{\left({x}^{{2}}-{4}\right)}\) since \(\displaystyle{a}^{{2}}\)

\(\displaystyle-{b}^{{2}}={\left({a}+{b}\right)}{\left({a}-{b}\right)}\)

\(\displaystyle={3}{\left({x}^{{2}}+{4}\right)}{\left({x}^{{2}}-{\left({2}\right)}^{{2}}\right)}\)

\(\displaystyle={3}{\left({x}^{{2}}+{4}\right)}{\left({x}+{2}\right)}{\left({x}-{2}\right)}\)

\(\displaystyle{3}{x}^{{4}}-{48}\)

To factorize the polynomials:

\(\displaystyle{3}{x}^{{4}}-{48}={3}{\left({x}^{{4}}-{16}\right)}\)

\(\displaystyle={3}{\left({\left({x}^{{2}}\right)}^{{2}}-{\left({4}\right)}^{{2}}\right)}\)

\(\displaystyle={3}{\left({x}^{{2}}+{4}\right)}{\left({x}^{{2}}-{4}\right)}\) since \(\displaystyle{a}^{{2}}\)

\(\displaystyle-{b}^{{2}}={\left({a}+{b}\right)}{\left({a}-{b}\right)}\)

\(\displaystyle={3}{\left({x}^{{2}}+{4}\right)}{\left({x}^{{2}}-{\left({2}\right)}^{{2}}\right)}\)

\(\displaystyle={3}{\left({x}^{{2}}+{4}\right)}{\left({x}+{2}\right)}{\left({x}-{2}\right)}\)