1Cademy - Subtracting $$\frac{n^2}{n-10}

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How to Add and Subtract Rational Expressions with a Common Denominator

Example

Subtracting $\frac{n^2}{n-10} - \frac{100}{n-10}$

Subtract two rational expressions that share the denominator $n - 10$ , then factor and simplify:

$\frac{n^2}{n - 10} - \frac{100}{n - 10}$

Step 1 — Subtract the numerators over the common denominator. Since both fractions share $n - 10$ as their denominator, subtract the numerators and place the difference over the common denominator:

$\frac{n^2 - 100}{n - 10}$

Step 2 — Factor the numerator. The numerator $n^2 - 100$ is a difference of squares, since $n^2 = n^2$ and $100 = 10^2$ . Apply the pattern $a^2 - b^2 = (a - b)(a + b)$ :

$\frac{(n - 10)(n + 10)}{n - 10}$

Step 3 — Simplify by dividing out the common factor. Cancel the shared factor $(n - 10)$ from the numerator and denominator:

$n + 10$

This example demonstrates all three steps of the rational expression subtraction procedure: subtracting the numerators, factoring the resulting polynomial numerator (here using the difference of squares pattern), and canceling the common factor with the denominator. The difference of squares $n^2 - 100 = (n - 10)(n + 10)$ produces a factor that matches the denominator, allowing the expression to simplify to a binomial.

Updated 2026-04-21

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OpenStax Elementary Algebra

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