Rational Zeros Calculator
Result
Rational Root Theorem
The Rational Root Theorem states that if a polynomial has a rational zero p/q, then:
- p is a factor of the constant term
- q is a factor of the leading coefficient
For a polynomial:
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Possible rational zeros are of the form ±p/q, where:
- p is a factor of a₀ (constant term)
- q is a factor of aₙ (leading coefficient)
Example Problems
Example 1: x³ - 2x² - 5x + 6
Step 1: Identify coefficients
- a₃ = 1 (leading coefficient)
- a₀ = 6 (constant term)
Step 2: Find factors
- Factors of 6: ±1, ±2, ±3, ±6
- Factors of 1: ±1
Step 3: Possible rational zeros
±1, ±2, ±3, ±6
Actual zeros: x = 1, x = 2, x = -3
Example 2: 2x² + 5x - 3
Step 1: Identify coefficients
- a₂ = 2 (leading coefficient)
- a₀ = -3 (constant term)
Step 2: Find factors
- Factors of -3: ±1, ±3
- Factors of 2: ±1, ±2
Step 3: Possible rational zeros
±1, ±1/2, ±3, ±3/2
Actual zeros: x = 1/2, x = -3
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