Factor Completely Calculator

Result

Enter a polynomial and press Factor Now to see results.

Example Data

Expression	Variable	Expected Factorization
6x^2 + 9x - 15	x	3(2x - 1)(x + 5)
x^2 - 9	x	(x - 3)(x + 3)
x^3 - 6x^2 + 11x - 6	x	(x - 1)(x - 2)(x - 3)
x^3 + 8	x	(x + 2)(x^2 - 2x + 4)

Click Use to load the example into the form.

Formulas Used

Greatest Common Factor (GCF): factor out numeric and variable powers common to all terms.
Difference of Squares: a^2 - b^2 = (a - b)(a + b).
Perfect Square Trinomial: a^2 ± 2ab + b^2 = (a ± b)^2.
Sum/Difference of Cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2), a^3 - b^3 = (a - b)(a^2 + ab - b^2).
Quadratic Factoring: find integers m,p,n,q with mp=a, nq=c, mq+np=b.
Rational Root Theorem: candidates are ±p/q with p|a_0, q|a_n. Exact division by (q x - p) uses integer recurrences.
Synthetic/Long Division: used to reduce degree after finding a root or factor.

Scope: one variable, integer coefficients; optional rational roots. Expressions should be entered in expanded form.

How to Use This Calculator

Enter your polynomial in expanded form, e.g., 6x^2 + 9x - 15.
Choose the variable symbol, usually x.
Optionally enable rational root search to allow denominators.
Click Factor Now to compute the complete factorization.
Review canonical form, degree, factorization, and the step-by-step breakdown.
Export your results using Download PDF or CSV.
Explore the example table to learn patterns and verify results.

how to get trinomials factor completely using calculator?

For a trinomial ax^2 + bx + c with integer coefficients, use these steps in the calculator:

Enter the trinomial in expanded form and choose the variable.
Factor out any GCF, numeric and variable powers, to simplify first.
If a = 1 (monic), find integers u, v with u·v = c and u + v = b, giving (x + u)(x + v).
If a ≠ 1, use the ac-method: find m, n with m·n = a·c and m + n = b, then split the middle term and factor by grouping.
Enable “rational root search” if integer pairing fails; this tries factors with denominators via the Rational Root Theorem.
Result review: the tool shows the complete factorization or states that the quadratic is irreducible over the integers.

Worked examples

x^2 + 5x + 6 → find u=2, v=3 → (x + 2)(x + 3).
6x^2 + 11x + 3 → a·c = 18; choose m=2, n=9; split: 6x^2 + 2x + 9x + 3 → grouping → (3x + 1)(2x + 3).
If the discriminant Δ = b^2 - 4ac is not a perfect square, integer factoring may fail; the calculator will report it as irreducible over integers.

what factors are considered when calculating completion rate?

Completion rate is generally computed as Completed ÷ Assigned × 100%. The exact value depends on how each component is defined and measured.

Definition of “completed”: fully finished vs. partial credit, acceptance criteria met.
Scope & complexity: weighting simple vs. complex items, time-on-task differences.
Time window: day, sprint, month, rolling period affects numerator/denominator.
Exclusions: cancelled, out-of-scope, duplicate, or test items removed from totals.
Retries & rework: whether failed attempts count, and how reopens are handled.
Backlog carryover: items moved across periods—counted when started or finished.
Data source quality: tracking system completeness, missing timestamps, manual edits.
Weighting schemes: equal-weight items or weighted by effort, priority, or size.
Cutoff rules: deadlines, grace periods, late completions counted or excluded.

Formula templates

Basic: CR = (Completed Items / Total Assigned Items) × 100%

Weighted: CR_w = (Σ w_i·done_i / Σ w_i) × 100%

Tip: When reporting completion rate, always document inclusion/exclusion rules, the period, and any weighting so results are reproducible and comparable.

common patterns to factor completely

Use this quick checklist to recognize patterns before you apply general methods:

GCF first: always pull out numeric and variable common factors.
Difference of squares: a^2 - b^2 = (a - b)(a + b); e.g., x^2 - 25 = (x - 5)(x + 5).
Perfect square trinomials: a^2 ± 2ab + b^2 = (a ± b)^2; e.g., x^2 + 6x + 9 = (x + 3)^2.
Sum/difference of cubes: a^3 ± b^3 = (a ± b)(a^2 ∓ ab + b^2); e.g., x^3 + 8 = (x + 2)(x^2 - 2x + 4).
Quadratic (ac-method): for ax^2 + bx + c, find m,n with mn=ac and m+n=b; split and group.
Grouping: with four terms, group pairs to expose a common binomial.
Rational Root Theorem: try ±p/q where p|a_0, q|a_n; use synthetic division to reduce degree.
Multiplicity: repeated linear factors combine as powers, e.g., (x-2)^3.
Irreducible quadratics: when the discriminant Δ=b^2-4ac isn’t a perfect square over integers.

troubleshooting: when factoring fails or stops

Not fully expanded: ensure input is expanded, e.g., enter x^3 - 6x^2 + 11x - 6 not (x-1)(x-2)(x-3).
No integer roots: enable the “rational root search” toggle to test p/q candidates.
Irreducible over integers: some quadratics won’t factor with integers; the tool reports irreducible status.
GCF missed: if results look messy, factor out a GCF first to simplify coefficients.
Variable mismatch: set the correct variable symbol (default is x).
Unsupported inputs: only one variable and integer coefficients; avoid decimals, fractions, or negative exponents.
Check discriminant: for ax^2+bx+c, if Δ is a perfect square, integer factoring is likely.
Verify signs carefully: small sign errors change roots; compare canonical form shown by the calculator.

Tip: After each successful factor, synthetic division reduces the degree, making the next factors easier to find.