How to Solve Polynomial Equations

To solve polynomial equations, pick a method—factor, formula, graph, or numeric—and test roots, then reduce until all solutions are found.

Solving equations built from powers of a variable looks tough until you sort the tools. This guide lays out the main playbook, shows how to pick the right move, and walks through worked problems from linear all the way to quartic. You’ll see where factoring shines, where formulas step in, and where graphing or calculators finish the job.

Solving Polynomial Equations Step By Step

Every non-constant one-variable polynomial has as many complex solutions as its degree, counted with repeats. That promise means a plan can always land a result. Start with a quick scan, choose a method, and peel the problem down until only simple pieces remain.

Methods At A Glance

Method	When It Works	Quick How-To
Inspection & Factoring	Small integer roots; common factors; patterns	Pull out GCF; spot (x+a)(x+b), squares, cubes; expand check
Quadratic Formula	Degree 2, any coefficients	Use x = [−b ± √(b² − 4ac)]/(2a); simplify radicals
Completing The Square	Quadratics that don’t factor cleanly	Move constant; add (b/2a)² both sides; take roots
Synthetic/Long Division	When a root is known or suspected	Divide to reduce degree; repeat on the quotient
Rational Root Hunt	Coefficients are integers	Test ±(factors of constant)/(factors of leading coeff.)
Graph/Numeric Solve	Messy higher degrees	Plot, bracket sign changes; use a solver to refine

Start With Simple Wins: Linear Pieces

If any factor is a first-degree term, set it to zero and solve right away. Try this, from (x−3)(x+2)=0 you read x=3 and x=−2. Remove solved factors to shorten what remains.

Quadratics: Three Reliable Paths

Try Factoring First

Rearrange to ax²+bx+c=0. If two integers multiply to ac and add to b, split the middle term or jump straight to (x+p)(x+q)=0. When that fizzles, move to a sure thing.

Use The Formula

The quadratic formula proof shows why x = [−b ± √(b² − 4ac)]/(2a) always lands roots. Compute the discriminant Δ=b²−4ac. If Δ>0, you get two real values; if Δ=0, one repeated real; if Δ<0, a complex pair.

Complete The Square

Divide through by a (if a≠0), shift c to the right, add (b/2a)² to both sides, then take square roots. This mirrors the formula and helps with vertex form in graph work.

Find Easy Starts With The Rational Root Rule

When coefficients are integers, any rational solution must look like p/q where p divides the constant term and q divides the leading coefficient. Test the small candidates first; one hit lets you divide and lower the degree.

Estimate How Many Real Roots

Count sign changes in the ordered coefficients to cap the number of positive real solutions; repeat with x→−x for negatives. This quick count narrows your search plan.

Peel Down With Synthetic Division

Once a root r is found, run synthetic division to factor out (x−r). The quotient drops the degree by one. Keep going until every factor is linear or quadratic. Solve each piece.

Handling Cubics And Quartics

Many third- and fourth-degree problems break after one rational hit. Try the integer list from the rule, divide once, then tackle the smaller quadratic or cubic that remains. Special shapes also help: a depressed cubic (no x² term) or a biquadratic (only even powers) reduces with a simple change such as y=x².

Pick The Right Tool For Each Degree

Graph views and calculators speed work once simple tricks stall. A plot shows where the curve crosses the axis and where it just kisses it. From there, a bisection or Newton step locks onto a decimal root, and division finishes the factorization.

Worked Problem A: Cubic With A Rational Start

Problem. Solve x³−4x²−x+4=0.

Scan. Try integer candidates ±1, ±2, ±4. Plugging x=1 gives 0, so x=1 is a solution.

Reduce. Divide by (x−1) to get x²−3x−4. Factor to (x−4)(x+1).

Finish. The solutions are x=1, x=4, and x=−1.

Worked Problem B: Quadratic With No Rational Roots

Problem. Solve 3x²+2x+5=0.

Discriminant. Δ=2²−4·3·5=4−60=−56.

Roots. x = [−2 ± √(−56)]/6 = (−2 ± i√56)/6 = (−1 ± i√14)/3.

Worked Problem C: Biquadratic Pattern

Problem. Solve x⁴−5x²+6=0.

Substitute. Let y=x². Then y²−5y+6=0, so (y−2)(y−3)=0.

Back-substitute. y=2 or y=3. That gives x=±√2 and x=±√3.

Why A Solution Always Exists

The Fundamental Theorem of Algebra guarantees that a non-constant single-variable polynomial has exactly as many complex solutions as its degree, counting repeats. That promise lets you factor completely over the complex numbers once you’ve found the real pieces.

When Decimals Are Fine

Textbook sets often land clean values. Real data seldom does. After you rule out simple candidates, switch to numeric search. Two safe, quick moves are:

Bisection. Pick an interval where the sign flips, halve it again and again to tighten the root.
Newton’s Method. Start near a crossing, use x_new=x−f(x)/f′(x). This converges fast near a simple root.

Once a decimal is good enough, divide by the matching factor and keep going.

Read The Discriminant And Multiplicity

Two quick diagnostics speed decisions in the middle of a solve.

Discriminant Signals For Quadratics

Δ>0: two real crossings.
Δ=0: one repeated real root (the graph touches the axis).
Δ<0: two complex results.

Multiplicity And The Graph

If (x−r) appears k times, r has multiplicity k. Odd k gives a cross; even k gives a touch. This helps confirm your factorization from a graph.

Common Pitfalls And Fixes

Symptom	Likely Cause	What To Do
No rational hit	Constant or leading coefficient has many factors	Trim list; try ±1, ±prime factors first; switch to graph/numeric
Weird decimals	Root is irrational or complex	Keep exact radicals when Δ isn’t a square; show complex pair
Stuck after one factor	Quotient still hard	Check for a quadratic; apply formula or complete the square
Sign count mismatch	Skipped a zero term in ordering	Write all powers in order; include zeros as placeholders
Newton bounces	Starting guess too far or flat slope	Bracket first with bisection; retry Newton near the crossing

Step-By-Step Playbook

Clean The Form. Move all terms to one side and combine like terms.
Check For GCF. Factor out any common number or variable power.
Scan Signs. Use sign changes for a quick count of possible positive and negative real roots.
List Easy Candidates. Write the small rational options p/q and test with substitution.
Divide On A Hit. Use synthetic division to drop the degree and repeat.
Handle Quadratics. Use the formula or square-completion on the last pieces.
Use Graphs Or A Solver. When the list runs dry, bracket crossings and compute decimals.
State All Solutions. Include complex values when they appear.

Worked Problem D: Mixed Factoring And Formula

Problem. Solve 2x³−3x²−8x+12=0.

Try Small Candidates. Test x=1: 2−3−8+12=3 (not zero). Test x=2: 16−12−16+12=0, so x=2 works.

Divide. Synthetic division by (x−2) gives 2x²+x−6. Solve that with the formula: x=[−1±√(1+48)]/4=[−1±7]/4.

Finish. The full set is x=2, x=3/2, and x=−2.

Worked Problem E: Decimal Roots By Search

Problem. Solve x³−x−1=0.

No Easy Rationals. The candidate list ±1 gives f(1)=−1 and f(−1)=1, so a crossing lies between. Bisection locks a value near 1.3247. Divide by (x−1.3247) and polish the remaining quadratic with a calculator to get the complex pair.

Reliable Patterns To Spot

Difference Of Squares. a²−b²=(a−b)(a+b).
Sum/Difference Of Cubes. a³±b³=(a±b)(a²∓ab+b²).
Biquadratic. Only even powers? Let y=x².
Symmetry. Terms in xⁿ and xⁿ⁻² hint at a y=x−1/x trick, but in school work the earlier patterns usually land faster.

Clean Answers That Teachers Like

Write roots clearly, group exact forms before decimals, and show the factorization. If you used a decimal step, say how you got it. A tidy final line might look like “(x−2)(x−3/2)(x+2)=0, so x=2, 3/2, −2.”

When you present complex results, pair them: “x=(−1±i√14)/3.” This signals that the work is complete.

Speed Checks And Finishing Touches

Plug Back In. After each root is found, test it in the original left-hand side. This catches slip-ups from sign errors or rushed arithmetic. When numbers are messy, keep an exact form during the check; cancel common factors before multiplying.

Mind Units And Context. In word problems, only some roots may fit the setting. A negative length, for instance, doesn’t make sense. State all algebraic solutions, then state which ones fit the story so your grader can see both parts.

Scale Before You Start. If every term shares a factor, divide it out to shrink the numbers. This keeps fractions under control when you move to the formula or synthetic steps.

Write Coefficients In Order. Missing powers need zeros. For a quartic like 2x⁴−5x²+3, record it as 2, 0, −5, 0, 3 before you run synthetic division. That single habit avoids many arithmetic traps.

Pick A Smart Start For Newton. Use the graph to pick a point near a crossing, not on a flat spot. If the slope is tiny, one jump can overshoot. Bracket with bisection first when in doubt, then let Newton finish the job.