Question

Use a Special Factoring Formula to factor the expression

$27x^3+y^3$

Short answer

The solution$27x^3+y^3=\left(3x+y\right)\left[9x^2+y^2-3xy\right]$

Step-by-step solution

Step 1. Apply the cube of sum formula.

If A and B are any real numbers or algebraic expressions, then

$\begin{array}{l}{\left(A+B\right)}^3=A^3+B^3+3A^{2}B+3A{B}^2\\ \Rightarrow A^3+B^3={\left(A+B\right)}^3-3AB(A+B)\\ \Rightarrow A^3+B^3=(A+B)\left[{\left(A+B\right)}^2-3AB\right]\\ \Rightarrow A^3+B^3=(A+B)\left[A^2+B^2+2AB-3AB\right]\\ \Rightarrow A^3+B^3=(A+B)\left[A^2+B^2-AB\right]\end{array}$

Step 2. Analyze the given expression.

Given$27x^3+y^3$

Here

$\begin{array}{l}{A}^3=27x^3,B^3=y^3\\ \Rightarrow A=3x,B=y\end{array}$

Step 3. Solve the expression.

Substitute the values in the Factoring formula to get:

$\begin{array}{l}27x^3+y^3={\left(3x\right)}^3+y^3\\ \Rightarrow 27x^3+y^3=\left(3x+y\right)\left[{\left(3x\right)}^2+y^2-\left(3x\right)y\right]\\ \Rightarrow 27x^3+y^3=\left(3x+y\right)\left[9x^2+y^2-3xy\right]\end{array}$