1Cademy - Factoring $$100

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Difference of Squares Pattern

Example

Factoring $100 - h^2$

Factor $100 - h^2$ by recognizing it as a difference of squares even though the constant term appears first — a "backwards" arrangement.

Step 1 — Is this a difference of squares? Yes. The expression is still a difference of two perfect squares, just written with the constant before the variable:

$100 = (10)^2$ , so $a = 10$ .
$h^2 = (h)^2$ , so $b = h$ .

Rewrite to show the squared structure: $(10)^2 - (h)^2$ .

Step 2 — Factor as the product of conjugates. Apply the difference of squares pattern with $a = 10$ and $b = h$ :

$100 - h^2 = (10 - h)(10 + h)$

Step 3 — Check by multiplying: $(10 - h)(10 + h) = 100 + 10h - 10h - h^2 = 100 - h^2$ ✓

The factored form is $(10 - h)(10 + h)$ .

Important caution: Do not rearrange the original expression to $h^2 - 100$ before factoring. Factoring $h^2 - 100$ gives $(h - 10)(h + 10)$ , which is a different — though related — result. The order of the original terms determines the order of the terms in the conjugate factors. To see the difference, compare: factoring $100 - h^2$ yields $(10 - h)(10 + h)$ , while factoring $h^2 - 100$ yields $(h - 10)(h + 10)$ .

Updated 2026-04-21

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OpenStax Elementary Algebra

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