A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Factoring polynomials involves breaking up a polynomial into simpler terms (the factors) such that when the terms are multiplied together they equal the original polynomial.

The given expression is \(\displaystyle{1}-{27}{a}^{{3}}\). Factor the given expression by simplifying and then using sum of two cubes pattern as follows:

\(\displaystyle{1}-{27}{a}^{{3}}={1}^{{3}}-{3}^{{3}}{a}^{{3}}\)

\(\displaystyle{\left({1}\right)}^{{3}}+{\left(-{3}{a}\right)}^{{3}}\)

\(\displaystyle={\left({1}+{\left(-{3}{a}\right)}\right)}{\left({1}^{{2}}-{\left({1}\right)}{\left(-{3}{a}\right)}+{\left(-{3}{a}\right)}^{{2}}\right)}\)

Use sum of two cubes pattern \(\displaystyle{a}^{{3}}+{b}^{{3}}={\left({a}+{b}\right)}{\left({a}^{{2}}-{a}{b}+{b}^{{2}}\right)}\)

\(\displaystyle={\left({1}-{3}{a}\right)}{\left({1}+{3}{a}+{9}{a}^{{2}}\right)}\)

\(\displaystyle={\left({1}-{3}{a}\right)}{\left({9}{a}^{{2}}+{3}{a}+{1}\right)}\) - Factored form.

Hence, the factored form of given expression is equal to \(\displaystyle{1}-{27}{a}^{{3}}={\left({1}-{3}{a}\right)}{\left({9}{a}^{{2}}+{3}{a}+{1}\right)}\).

The given expression is \(\displaystyle{1}-{27}{a}^{{3}}\). Factor the given expression by simplifying and then using sum of two cubes pattern as follows:

\(\displaystyle{1}-{27}{a}^{{3}}={1}^{{3}}-{3}^{{3}}{a}^{{3}}\)

\(\displaystyle{\left({1}\right)}^{{3}}+{\left(-{3}{a}\right)}^{{3}}\)

\(\displaystyle={\left({1}+{\left(-{3}{a}\right)}\right)}{\left({1}^{{2}}-{\left({1}\right)}{\left(-{3}{a}\right)}+{\left(-{3}{a}\right)}^{{2}}\right)}\)

Use sum of two cubes pattern \(\displaystyle{a}^{{3}}+{b}^{{3}}={\left({a}+{b}\right)}{\left({a}^{{2}}-{a}{b}+{b}^{{2}}\right)}\)

\(\displaystyle={\left({1}-{3}{a}\right)}{\left({1}+{3}{a}+{9}{a}^{{2}}\right)}\)

\(\displaystyle={\left({1}-{3}{a}\right)}{\left({9}{a}^{{2}}+{3}{a}+{1}\right)}\) - Factored form.

Hence, the factored form of given expression is equal to \(\displaystyle{1}-{27}{a}^{{3}}={\left({1}-{3}{a}\right)}{\left({9}{a}^{{2}}+{3}{a}+{1}\right)}\).