Factor $27 h^3-125 d^3$ completely.
The expression to be factored is \(27h^3 - 125d^3\). This expression is a difference of two cubes, which follows the algebraic pattern \(a^3 - b^3\). The identity for the difference of two cubes is given by:
\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]
To use the formula, we must identify \(a\) and \(b\) such that \(a^3 = 27h^3\) and \(b^3 = 125d^3\).
Taking the cube root of each term:
\[a = \sqrt[3]{27h^3} = 3h\]
\[b = \sqrt[3]{125d^3} = 5d\]
Now, we substitute \(a = 3h\) and \(b = 5d\) into the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\):
\[27h^3 - 125d^3 = (3h - 5d)((3h)^2 + (3h)(5d) + (5d)^2)\]
Calculating each component within the second factor:
\((3h)^2 = 9h^2\)
\((3h)(5d) = 15hd\)
\((5d)^2 = 25d^2\)
Substituting these values back into the expression yields:
\[27h^3 - 125d^3 = (3h - 5d)(9h^2 + 15hd + 25d^2)\]
The expression is now completely factored. The quadratic term \(9h^2 + 15hd + 25d^2\) has a negative discriminant, which confirms it cannot be factored further using real coefficients. Thus, the final factored form is \((3h - 5d)(9h^2 + 15hd + 25d^2)\).
In engineering and physics, factoring difference of cubes is often useful when simplifying complex volumetric or power-related formulas. For instance, if you are designing a mechanical component with a cubic volume defined by \(27h^3\) and you need to subtract an internal void of volume \(125d^3\), factoring this into \((3h-5d)(9h^2+15hd+25d^2)\) can reveal specific geometric relationships between the dimensions \(h\) and \(d\), allowing you to better optimize the part's material usage or stress distribution.