Question

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
$(x+4{)}^3-27x^3$

Short answer

The factored form of the given expression is $2(2-x)(13x^2+20x+16)$.

Step-by-step solution

Step 1. Given information.

The given expression is:
$(x+4{)}^3-27x^3$

Step 2. Determine the factored form.

The given expression can be written as:
$\begin{array}{rcl}(x+4{)}^3-27x^3& =& (x+4{)}^3-{\left(3x\right)}^3\\ & =& (x+4-3x)({(x+4)}^2+(x+4\left)\right(3x)+{\left(3x\right)}^2)\left[\because a^3-b^3=(a-b)(a^2+ab+b^2)\right]\\ & =& (4-2x)(x^2+8x+16+3x^2+12x+9x^2)\left[\because {(a+b)}^2=a^2+2ab+b^2\right]\\ & =& 2(2-x)(13x^2+20x+16)\end{array}$

Step 3. Conclusion.

The factored form of the given expression is $2(2-x)(13x^2+20x+16)$.