Subtracting
Subtract two rational expressions whose numerators are both trinomials and that share the common denominator , then factor both the numerator and denominator to simplify:
Step 1 — Subtract the numerators over the common denominator. Place the second numerator in parentheses and write the difference over the shared denominator:
Step 2 — Distribute the negative sign in the numerator. Multiply each term inside the parentheses by , changing to , to , and to :
Step 3 — Combine like terms in the numerator. Group the -terms: . Group the -terms: . Group the constants: :
Step 4 — Factor both the numerator and the denominator. The numerator factors as , since and . The denominator factors as , since and :
Step 5 — Simplify by removing the common factor. Cancel the shared factor from the numerator and denominator:
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 would remain hidden.
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