Example

Subtracting 5x2−7x+3x2−3x−18−4x2+x−9x2−3x−18\frac{5x^2-7x+3}{x^2-3x-18} - \frac{4x^2+x-9}{x^2-3x-18}

Subtract two rational expressions whose numerators are both trinomials and that share the common denominator x2−3x−18x^2 - 3x - 18, then factor both the numerator and denominator to simplify:

5x2−7x+3x2−3x−18−4x2+x−9x2−3x−18\frac{5x^2 - 7x + 3}{x^2 - 3x - 18} - \frac{4x^2 + x - 9}{x^2 - 3x - 18}

Step 1 — Subtract the numerators over the common denominator. Place the second numerator in parentheses and write the difference over the shared denominator:

5x2−7x+3−(4x2+x−9)x2−3x−18\frac{5x^2 - 7x + 3 - (4x^2 + x - 9)}{x^2 - 3x - 18}

Step 2 — Distribute the negative sign in the numerator. Multiply each term inside the parentheses by −1-1, changing +4x2+4x^2 to −4x2-4x^2, +x+x to −x-x, and −9-9 to +9+9:

5x2−7x+3−4x2−x+9x2−3x−18\frac{5x^2 - 7x + 3 - 4x^2 - x + 9}{x^2 - 3x - 18}

Step 3 — Combine like terms in the numerator. Group the x2x^2-terms: 5x2−4x2=x25x^2 - 4x^2 = x^2. Group the xx-terms: −7x−x=−8x-7x - x = -8x. Group the constants: 3+9=123 + 9 = 12:

x2−8x+12x2−3x−18\frac{x^2 - 8x + 12}{x^2 - 3x - 18}

Step 4 — Factor both the numerator and the denominator. The numerator x2−8x+12x^2 - 8x + 12 factors as (x−2)(x−6)(x - 2)(x - 6), since (−2)+(−6)=−8(-2) + (-6) = -8 and (−2)(−6)=12(-2)(-6) = 12. The denominator x2−3x−18x^2 - 3x - 18 factors as (x+3)(x−6)(x + 3)(x - 6), since 3+(−6)=−33 + (-6) = -3 and 3⋅(−6)=−183 \cdot (-6) = -18:

(x−2)(x−6)(x+3)(x−6)\frac{(x - 2)(x - 6)}{(x + 3)(x - 6)}

Step 5 — Simplify by removing the common factor. Cancel the shared factor (x−6)(x - 6) from the numerator and denominator:

x−2x+3\frac{x - 2}{x + 3}

This example extends the earlier subtraction examples in two important ways. First, both numerators are trinomials rather than a monomial and a binomial, so distributing the negative sign affects three terms instead of one or two. Second, the denominator itself must also be factored — not just the combined numerator — before the common factor can be identified and canceled. Without factoring the denominator, the shared factor (x−6)(x - 6) would remain hidden.

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Updated 2026-05-14

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