Formula for the sum and difference of the cubes of two numbers - $(a\pm b)(a^2\mp ab+b^2)$
We continue with the next and last of the abbreviated multiplication formulas $(a\pm b)(a^2\mp ab+b^2)=a^3\pm b^3$. We will look at some problems to show some applications of it. Problem 1 Perform the multiplication $(3-x)(9+3x+x^2).$ Solution: Notice that given the expression $(3-x)(9+3x+x^2)$, we can write it in the form $(3-x)(3^2+3x+x^2).$ We will apply the formula $(a-b)(a^2+ab+b^2)$, replacing $a$ with $3$ and $b$ with $x$, so we get $(3-x)(3^2+3x+x^2)=3^3-x^3.$ Problem 2 Perform the multiplication $(3t+2)(9t^2-6t+4).$ Solution: Given an expression, we can write it in the form $(3t+2)[(3t)^2-3t.2+2^2]$. Now it is easy to see that we can apply the formula $(a+b)(a^2-ab+b^2)$, where $a=3t$ and $b=2$, so we get $(3t+2)[(3t)^2-3t.2+2^2]=(3t)^3+2^3=27t^3+8.$ Problem 3 Simplify the expression $(x-2)(x^2+2x+4)-x(x-2)(x+2)-4(x-2).$ Solution: Apply the formulas $(a-b)(a^2+ab+b^2)=a^3-b^3$ and $(a-b)(a+b)=a^2-b^2$, hence $(x-2)(x^2+2x+4)-x(x-2)(x+2)-4(x-2)=x^3-2^3-x(x^2-4)-4x+8=x^3-8...