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Example of Dividing Rational Expressions

To divide rational expressions, rewrite the division as multiplication by the reciprocal of the second fraction, factor all polynomials completely, and simplify by dividing out common factors.

For example, to simplify p3+q32p2+2pq+2q2÷p2−q26\frac{p^3 + q^3}{2p^2 + 2pq + 2q^2} \div \frac{p^2 - q^2}{6}:

Step 1. Rewrite the division as multiplication by the reciprocal of the second fraction: p3+q32p2+2pq+2q2⋅6p2−q2\frac{p^3 + q^3}{2p^2 + 2pq + 2q^2} \cdot \frac{6}{p^2 - q^2}

Step 2. Factor the numerators and denominators completely. The first numerator is a sum of cubes: p3+q3=(p+q)(p2−pq+q2)p^3 + q^3 = (p + q)(p^2 - pq + q^2). The first denominator has a common numerical factor of 22: 2(p2+pq+q2)2(p^2 + pq + q^2). The second numerator is 6=2⋅36 = 2 \cdot 3. The second denominator is a difference of squares: p2−q2=(p−q)(p+q)p^2 - q^2 = (p - q)(p + q). Substituting these yields: (p+q)(p2−pq+q2)2(p2+pq+q2)⋅2⋅3(p−q)(p+q)\frac{(p + q)(p^2 - pq + q^2)}{2(p^2 + pq + q^2)} \cdot \frac{2 \cdot 3}{(p - q)(p + q)}

Step 3. Multiply the numerators and denominators together: (p+q)(p2−pq+q2)⋅2⋅32(p2+pq+q2)(p−q)(p+q)\frac{(p + q)(p^2 - pq + q^2) \cdot 2 \cdot 3}{2(p^2 + pq + q^2)(p - q)(p + q)}

Step 4. Simplify by dividing out the common factors of (p+q)(p + q) and 22: 3(p2−pq+q2)(p−q)(p2+pq+q2)\frac{3(p^2 - pq + q^2)}{(p - q)(p^2 + pq + q^2)}

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Updated 2026-05-25

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References

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OpenStax

Intermediate Algebra @ OpenStax

Ch.7 Rational Expressions and Functions - Intermediate Algebra @ OpenStax

Algebra

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