Q. 209 - Intermediate Algebra

Question

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
$(x+3{)}^3+8x^3$

Step-by-Step Solution

Answer

The factored form of the given expression is $9(x+1)(x^2+3)$.

Step 1. Given information.

The given expression is:
$(x+3{)}^3+8x^3$

Step 2. Determine the factored form.

The given expression can be written as:
$\begin{array}{rcl}(x+3{)}^3+8x^3& =& (x+3{)}^3+{\left(2x\right)}^3 \\ & =& (x+3+2x)({(x+3)}^2-(x+3\left)\right(2x)+{\left(2x\right)}^2) \left[\because a^3+b^3=(a+b)(a^2-ab+b^2)\right]\\ & =& (3x+3)(x^2+6x+9-2x^2-6x+4x^2) \left[\because {(a+b)}^2=a^2+2ab+b^2\right]\\ & =& 3(x+1)(3x^2+9)\\ & =& 9(x+1)(x^2+3)\end{array}$

Step 3. Conclusion.

The factored form of the given expression is $9(x+1)(x^2+3)$.