Factor $27 h^3-125 d^3$ completely.

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03-11-2026
Mathematics

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Factor $27 h^3-125 d^3$ completely.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T08:17:05-05:00

Identify the expression as a difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
Recognize $a = 3h$ and $b = 5d$.
Substitute these into the formula to get $(3h - 5d)((3h)^2 + (3h)(5d) + (5d)^2)$.
Simplify to obtain the final factored form: $\boxed{(3h - 5d)(9h^2 + 15hd + 25d^2)}$.

Explanation

Identifying the Algebraic Structure

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:
\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]

Determining the Terms for the Formula

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:
\[a = \sqrt[3]{27h^3} = 3h\]
\[b = \sqrt[3]{125d^3} = 5d\]

Applying the Formula and Simplifying

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:
\[27h^3 - 125d^3 = (3h - 5d)(9h^2 + 15hd + 25d^2)\]

Conclusion and Final Result

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 $(3h - 5d)(9h^2 + 15hd + 25d^2)$.

Examples

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.