Polynomials & Rational Expressions
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Rational Expressions & Partial Fractions

📐 Polynomials & Rational Expressions
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Rational Expressions & Partial Fractions

Simplify rational expressions and decompose fractions for integration prep.

A rational expression is a ratio of two polynomials p(x)/q(x) where q(x) ≠ 0. We simplify by factoring and canceling common factors. Partial fraction decomposition reverses this process, breaking a complex rational expression into simpler pieces.

x2−4x2−x−6=(x−2)(x+2)(x−3)(x+2)=x−2x−3(x≠−2)\frac{x^2 - 4}{x^2 - x - 6} = \frac{(x-2)(x+2)}{(x-3)(x+2)} = \frac{x-2}{x-3} \quad (x \neq -2)

Simplifying Rational Expressions

1Completely factor the numerator and denominator.
2Cancel common factors (note the excluded values where q = 0).
3State the domain restrictions explicitly.

Partial fraction decomposition expresses a rational function as a sum of simpler fractions. For distinct linear factors: p(x)/[(x−a)(x−b)] = A/(x−a) + B/(x−b).

3x+5(x+1)(x−2)=Ax+1+Bx−2\frac{3x+5}{(x+1)(x-2)} = \frac{A}{x+1} + \frac{B}{x-2}
đŸ”ĸFinding A and B
Multiply both sides by (x+1)(x−2): 3x+5 = A(x−2) + B(x+1). Set x=2: 11 = 3B → B = 11/3. Set x=−1: 2 = −3A → A = −2/3.
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Remember This!

The "cover-up method": to find A/(x−a), cover up (x−a) in the denominator of the original fraction and evaluate the remaining expression at x = a.

✏ī¸ Try It!

Simplify: (x² − 9) / (x² + x − 6)

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