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Solving Polynomial & Rational Equations
AP Precalculus • Unit 1
Solving Polynomial & Rational Equations
Learn systematic methods to solve polynomial and rational equations, including factoring, zero-product principle, and eliminating extraneous solutions.
Lesson Objectives
- Solve polynomial equations using factoring
- Apply the zero-product property
- Solve rational equations correctly
- Identify and reject extraneous solutions
1. Solving Polynomial Equations
To solve a polynomial equation, we rewrite it in the form:
Polynomial = 0
Then we factor and apply the zero-product principle.
Zero-Product Principle:
If ab = 0, then a = 0 or b = 0.
If ab = 0, then a = 0 or b = 0.
2. Polynomial Equation Examples
Example 1:
Solve: x² − 9 = 0
Solve: x² − 9 = 0
(x − 3)(x + 3) = 0
x = 3 or x = −3
Example 2:
Solve: x³ − 4x = 0
Solve: x³ − 4x = 0
x(x² − 4) = 0
x = 0, 2, −2
Example 3 (AP-style):
Solve: 2x³ − 8x² = 0
Solve: 2x³ − 8x² = 0
2x²(x − 4) = 0
x = 0 (double root), x = 4
3. Solving Rational Equations
A rational equation contains fractions with variables in the denominator.
Strategy:
- Identify restrictions (values that make denominators zero)
- Multiply both sides by the LCD
- Solve the resulting polynomial equation
- Check for extraneous solutions
4. Rational Equation Examples
Example 4:
Solve: 1/(x − 2) = 3
Solve: 1/(x − 2) = 3
Multiply both sides by (x − 2):
1 = 3(x − 2)
x = 7/3
Example 5:
Solve: (x + 1)/(x − 1) = 2
Solve: (x + 1)/(x − 1) = 2
x + 1 = 2(x − 1)
x = 3
Example 6 (Extraneous Solution):
Solve: (x − 2)/(x − 2) = 1
Solve: (x − 2)/(x − 2) = 1
Restriction: x ≠ 2
Algebra gives x = 2 ❌ (reject)
No solution
5. Key AP Exam Notes
- Always state restrictions when solving rational equations
- Check all solutions in the original equation
- Extraneous solutions are common AP traps
- Multiplicity appears as repeated solutions
Practice Questions
- Solve: x² − 5x = 0
- Solve: x³ − x² − 6x = 0
- Solve: (x + 2)/(x − 3) = 1
- Solve: (x − 1)/(x − 1) = 2
✅ Show Answer Key
- x = 0, 5
- x = 0, 3, −2
- x = 5
- No solution (extraneous)
© Aviate Learning – AP Precalculus