1Cademy - Subtracting $$\frac{m^2-6m}{m^2-1}

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Adding and Subtracting Rational Expressions Whose Denominators Are Opposites

Example

Subtracting $\frac{m^2-6m}{m^2-1} - \frac{3m+2}{1-m^2}$

Subtract two rational expressions whose denominators are opposites, then factor and simplify:

$\frac{m^2 - 6m}{m^2 - 1} - \frac{3m + 2}{1 - m^2}$

Step 1 — Recognize the opposite denominators. The denominators $m^2 - 1$ and $1 - m^2$ are opposites because $1 - m^2 = -(m^2 - 1)$ . Multiply the numerator and denominator of the second fraction by $-1$ :

$\frac{m^2 - 6m}{m^2 - 1} - \frac{(3m + 2)(-1)}{(1 - m^2)(-1)}$

Step 2 — Simplify the second fraction. Multiplying gives $\frac{-3m - 2}{m^2 - 1}$ . Both fractions now share the denominator $m^2 - 1$ :

$\frac{m^2 - 6m}{m^2 - 1} - \frac{-3m - 2}{m^2 - 1}$

Step 3 — Subtract the numerators over the common denominator:

$\frac{m^2 - 6m - (-3m - 2)}{m^2 - 1}$

Step 4 — Distribute the negative sign. Subtracting $(-3m - 2)$ reverses the signs: $-(-3m) = +3m$ and $-(-2) = +2$ :

$\frac{m^2 - 6m + 3m + 2}{m^2 - 1}$

Step 5 — Combine like terms. Group the $m$ -terms: $-6m + 3m = -3m$ :

$\frac{m^2 - 3m + 2}{m^2 - 1}$

Step 6 — Factor the numerator and denominator. The numerator $m^2 - 3m + 2$ factors as $(m - 1)(m - 2)$ , since $(-1) + (-2) = -3$ and $(-1)(-2) = 2$ . The denominator $m^2 - 1$ is a difference of squares: $(m - 1)(m + 1)$ :

$\frac{(m - 1)(m - 2)}{(m - 1)(m + 1)}$

Step 7 — Simplify by removing the common factor. Cancel the shared factor $(m - 1)$ :

$\frac{m - 2}{m + 1}$

This example combines the opposite-denominators technique with polynomial subtraction, trinomial factoring, and the difference of squares pattern. After converting the denominators to match, the subtraction and sign distribution produce a trinomial numerator that factors and shares a common binomial factor with the denominator, enabling further simplification.

Updated 2026-06-24

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