How to solve polynomial equations with the DuckieDai Polynomial Toolkit

Quick reference workflow

Which program do I start with?

What each program is for

P1RZERO — Rational Zero Finder

Enter the constant term and leading coefficient. P1RZERO lists possible positive and negative rational zeros from the Rational Zero Theorem. These are candidates, not guaranteed roots.

P2Q4 and P3Q3 — Zero Testers

Enter the polynomial coefficients and one candidate zero. The program performs synthetic division. A remainder of 0 means the tested value is an actual zero; synthetic division then lowers the polynomial degree by one and gives the next coefficients.

P4QUAD — Quadratic Solver

Enter a, b, and c for ax² + bx + c = 0. It handles two real roots, one repeated root, and complex roots. Decimal answers are useful checks, but exact radical form may be required for homework.

P5RAD — Radical Simplifier

Use it when a square root from the quadratic formula can be simplified. For example, sqrt(333) = 3*sqrt(37). Enter fractions where the individual program supports them.

Worked quartic example

Solve 2x⁴ − 11x³ + 18x² − 22x + 28 = 0

1. Use P1RZERO

Enter constant 28 and leading coefficient 2. Test candidates from its list in P2Q4. One successful candidate is 2.

2. Test 2 in P2Q4

Enter coefficients 2, -11, 18, -22, 28 and test zero 2. The remainder is 0, so 2 is a root. Keep it. The cubic quotient is 2x³ − 7x² + 4x − 14.

3. Carry the quotient into P3Q3

Enter 2, -7, 4, -14. Test 7/2. Its remainder is 0, so 7/2 is another root. Keep it. The quadratic quotient is 2x² + 4.

4. Finish with P4QUAD

Enter a=2, b=0, c=4. The result is x = -i*sqrt(2) and x = i*sqrt(2). The radical is already simplified.

Final solution set: 2, 7/2, -i*sqrt(2), i*sqrt(2).

Short cubic example

Solve 9x³ − 12x² − 8x − 1 = 0

Use P1RZERO → P3Q3 → P4QUAD → P5RAD. P1RZERO gives candidates; test -1/3 in P3Q3. The remainder is 0, so keep -1/3. Synthetic division gives 9x² − 15x − 3.

Enter 9, -15, -3 in P4QUAD. The quadratic formula gives (15 ± sqrt(333)) / 18. P5RAD shows sqrt(333) = 3*sqrt(37), so the roots simplify to (5 - sqrt(37)) / 6 and (5 + sqrt(37)) / 6.

Final solution set: -1/3, (5-sqrt(37))/6, (5+sqrt(37))/6.

Troubleshooting

The remainder is not 0

That candidate is not a root. Return to P1RZERO and test another possible rational zero. Recheck coefficient signs and make sure you entered every term, including zero coefficients.

I have reached a quadratic

Stop using synthetic division and enter its three coefficients in P4QUAD. The quadratic formula is the finishing method once the degree is 2.

My answer is decimal but homework wants exact form

Use the displayed discriminant and P5RAD to simplify its square root where possible. Keep fractions exact instead of rounding early.

FAQ

Does a possible rational zero always work?

No. P1RZERO gives candidates from the Rational Zero Theorem. P2Q4 or P3Q3 verifies each one with a remainder.

What does a remainder of 0 mean?

It means the tested number is an actual zero and the polynomial factors by that linear factor.

Why do I need to save roots as I go?

Each successful division removes one factor and lowers the degree. The quotient only describes the roots that are still left to find.