Cubic Factorization Calculator
Factor cubic polynomials ax³ + bx² + cx + d completely into linear and quadratic factors using the Rational Root Theorem, Remainder Theorem testing, and synthetic division matrices.
Polynomial Coefficients
P(x) = ax³ + bx² + cx + dThe Rational Root Theorem operates on integer coefficients. Enter integers for a, b, c, and d.
Factors of constant d = -6 are ±6. Factors of leading coefficient a = 1 are ±1.
Step-by-Step Rational Root & Synthetic Division Breakdown
1 Testing Rational Candidates (Remainder Theorem: P(r) = 0)
Substituting test candidates into P(x) to identify a zero remainder:
| Candidate (r) | Evaluation P(r) | Result | Conclusion |
|---|
2 Synthetic Division by Linear Factor
Dividing the cubic polynomial by (x - r) yields the depressed quadratic quotient:
3 Factoring the Quadratic Quotient
Factoring the quadratic quotient into two linear factors or determining irreducibility.
How to Factor a Cubic Polynomial Using the Rational Root Theorem
To factor a cubic polynomial ax³ + bx² + cx + d: 1) Identify all integer factors of the constant term d (p) and leading coefficient a (q). 2) Form candidate roots ±p/q. 3) Test candidates in P(x) until P(r) = 0. 4) Use synthetic division to divide P(x) by (x - r), yielding a quadratic quotient Ax² + Bx + C. 5) Factor the quadratic quotient into linear factors to obtain the complete factored form.
Rational Root Theorem: Formal Definition & Proof
In commutative algebra and polynomial arithmetic, the Rational Root Theorem (also known as the rational zero theorem) establishes a direct relationship between the rational solutions of an integer polynomial equation and its boundary coefficients.
Let P(x) = anxn + an−1xn−1 + … + a₁x + a₀, with an ≠ 0 and all coefficients in ℤ.
If r = p / q (in lowest terms, gcd(|p|, |q|) = 1) is a rational zero of P(x) = 0, then:
p | a₀ (p divides constant term a₀) and q | an (q divides leading coefficient an)
For a degree-3 cubic P(x) = ax³ + bx² + cx + d, every rational zero must be formed by dividing an integer divisor of d by an integer divisor of a.
Systematic Candidate Set Generation (± p / q)
Generating the candidate set involves four deterministic steps:
- List all positive integer factors of the constant term |d|: p ∈ {p_1, p_2, …}.
- List all positive integer factors of the leading coefficient |a|: q ∈ {q_1, q_2, …}.
- Construct every fraction p / q in lowest terms (&gcd;(p, q) = 1).
- Attach both positive and negative signs (±) to every fraction, eliminating duplicates.
Example: P(x) = 2x³ - 5x² - 4x + 3
Constant d = 3 ⇒ p ∈ {1, 3}
Leading a = 2 ⇒ q ∈ {1, 2}
Candidate Set: ±{1/1, 3/1, 1/2, 3/2} = ±{1, 3, 1/2, 3/2}
Remainder Theorem & Fast Root Verification
Rather than performing cumbersome polynomial long division for every candidate, we invoke the Polynomial Remainder Theorem:
If direct substitution evaluates to P(r) = 0, the remainder is identically zero, rigorously confirming that (x - r) divides P(x) without remainder.
Synthetic Division Matrix & Depressed Quadratic
Once a confirmed zero r is identified, synthetic division efficiently depresses the degree from 3 down to 2:
Root r | a b c d
| — r×a r×B r×C
--------------------------------------------------
a B=b+r×a C=c+r×B 0 (Remainder)
The bottom row entries a, B, C form the coefficients of the depressed quadratic quotient: Q(x) = ax² + Bx + C.
Factoring Quadratic Quotients & Irreducibility
After extracting the first linear factor (x - r), the remaining quadratic factor Q(x) = Ax² + Bx + C is evaluated using its discriminant Δ = B² - 4AC:
If the discriminant is non-negative, the quadratic factors into two linear terms: A(x - r₂)(x - r₃). If Δ is a perfect square, r₂ and r₃ are rational numbers.
If the discriminant is negative, the quadratic has no real linear factors and remains irreducible over the real numbers: (x - r₁)(Ax² + Bx + C).
Step-by-Step Worked Factoring Examples
Example 1: Factor 2x³ - 5x² - 4x + 3
1. Candidates: ±{1, 3, 1/2, 3/2}
2. Test P(3): 2(27) - 5(9) - 4(3) + 3 = 54 - 45 - 12 + 3 = 0 ⇒ x = 3 is a root!
3. Linear factor: (x - 3)
4. Synthetic Division by 3: Quotient = 2x² + x - 1
5. Factor Quadratic: 2x² + x - 1 = (2x - 1)(x + 1)
6. Complete Factored Form: (x - 3)(2x - 1)(x + 1)
Common Pitfalls & Diagnostic Checks
Candidates are strictly p / q = (factors of constant) / (factors of leading). Inverting to q / p generates invalid candidate ratios.
A root of r = 3 corresponds to the factor (x - 3), not (x + 3). Remember that P(r) = 0 ⇒ (x - r) = 0.
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