Rational Zeros Calculator

Your details

The highest power of x in your polynomial. Coefficients must be integers.
The coefficient of the x^n term. Must be a non-zero integer.
Integer coefficient. Use 0 if that term is missing.
Integer coefficient.
For a cubic this is the constant. For higher degrees this is the x^(n-3) term and more coefficient fields appear.
Rational zeros found
3

How many of the possible candidates are confirmed rational zeros

Actual rational zerosx = 1, x = 2, x = 3
Possible rational zeros (candidates)-6, -3, -2, -1, 1, 2, 3, 6
Factors of constant term (p)±1, ±2, ±3, ±6
Factors of leading coefficient (q)±1
Total candidates to test8
Rational zeros found3
Total candidates tested8

3 rational zeros confirmed: x = 1, x = 2, x = 3.

  • 3 rational roots were confirmed by synthetic division.
  • All roots of the polynomial are rational - you have the complete factorization.
  • The theorem tested 8 candidates derived from the factors of the constant term and leading coefficient.
  • Because the leading coefficient is 1 (monic polynomial), all candidates are integers, making testing especially fast.

Next stepYou can write the full factored form: multiply the leading coefficient by each (x - root) factor.

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