# Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following. 1-27a^3

Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following.
$$\displaystyle{1}-{27}{a}^{{3}}$$

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Nathanael Webber
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 $$\displaystyle{1}-{27}{a}^{{3}}={\left({1}-{3}{a}\right)}{\left({9}{a}^{{2}}+{3}{a}+{1}\right)}$$.