Polynomial Calculator

Include zeros for missing powers, e.g. 1, 0, 0, 0, -1 for x^4 - 1.
Brackets and powers are accepted, e.g. (x - 2)^2(x + 1). An "= 0" is optional.
Applies to the printed roots; the returned numbers are full precision.
All roots
1, 2, 3
Polynomial Root & Factor Calculator
Degree
3
Leading coefficient
1
Real roots (with multiplicity)
3
Distinct real roots
3
Non-real roots
0
Smallest real root
1
Largest real root
3
Exact rational roots
1, 2, 3
Factorisation over the reals
(x - 1)(x - 2)(x - 3)
Standard form
x^3 - 6x^2 + 11x - 6 = 0
Sum of roots
6
Product of roots
6
Value at 0 (constant term)
-6
Worked out from OpenStax, College Algebra 2e, Section 5.5: Zeros of Polynomial Functions (Rational Zero Theorem, Factor Theorem, Descartes’ Rule of Signs) and 2 other sources.

How this is worked out

6 steps
  1. Polynomial: x^3 - 6x^2 + 11x - 6 (degree 3).
  2. Rational Root Theorem: testing p/q with p dividing the constant term and q dividing the leading coefficient finds 1, 2, 3.
  3. Dividing those factors out leaves 1.
  4. Roots: 1, 2, 3.
  5. Factorisation over the reals: (x - 1)(x - 2)(x - 3).
  6. Vieta check: sum of roots = -a(n-1)/a(n) = 6, product = (-1)^3 a0/a(n) = 6.
Real rootstable
1, 2, 3
Root tabletable
indexrealimaginarydisplaytyperesidual
1101real0
2202real0
3303real0

Results are general information, not financial, tax, legal, medical or engineering advice.