Proving Polynomial Identities

We want to state which special factoring pattern, if any, the given polynomial function follows. $$\begin{array}{c}{x}^2-4\end{array}$$ Note that the exponent of $x$ is $2,$ and that $4$ can be written as $2^2.$ Moreover, we have subtraction separating the terms. Therefore, the given expression follows the pattern $\text{difference}$ of two $\text{squares}.$ $$\begin{array}{c}{x}^2-2^2\end{array}$$

We want to state which special factoring pattern, if any, the given polynomial function follows. $$\begin{array}{c}3x^3+5\end{array}$$ Neither $3x^3$ nor $5$ can be written as perfect squares or cubes. Therefore, the given expression is not a difference of two squares, a sum of two cubes, or a difference of two cubes. We also see the expression has only two terms. This means it is not a perfect square trinomial. Consequently, it does not follow any special factoring pattern.

We want to state which special factoring pattern, if any, the given polynomial function follows. $$\begin{array}{c}4x^2+25\end{array}$$ Note that the first term can be written as $(2x{)}^2,$ and that $25$ can be written as $5^2.$ However, there is addition separating the terms. Therefore, the given expression is not a difference of two squares, a sum of two cubes, or a difference of two cubes. Moreover, the expression is not a perfect square trinomial because it only has two terms. Consequently, it does not follow any special factoring pattern.

We want to state which special factoring pattern, if any, the given polynomial function follows. Note that the exponent of $x$ is $3,$ that $27$ can be written as $3^3,$ and that $1000$ can be written as $10^3.$ $$\begin{array}{c}27x^3+1000\iff (3x{)}^3+10^3\end{array}$$ We can see that both terms of the expression are perfect cubes. Moreover, the terms are separated by addition. Therefore, the given expression follows the pattern $\text{sum}$ of two $\text{cubes}.$ $$\begin{array}{c}(3x{)}^3+10^3\end{array}$$

We want to state which special factoring pattern, if any, the given polynomial function follows. Since it has three terms, we know it is not a difference of two squares, a sum of two cubes, or a difference of two cubes. $$\begin{array}{c}64x^3-x^2+1\end{array}$$ Furthermore, note that, although $64$ is a perfect square, $64x^3$ is not perfect square because the exponent of the $x\text{-}$variable is $3.$ This means, the expression is not a perfect square trinomial. Therefore, it does not follow any special factoring pattern.