Factor x^3-6x^2+11x-6
Factor polynomials completely: trinomials, difference of squares, sum/difference of cubes.
To factor x^3-6x^2+11x-6, look for patterns like grouping, difference of squares, or rational roots.
x3-6x2+11x-6
By the Rational Root Theorem, any rational root p/q must have p dividing the constant term and q dividing the leading coefficient:
Factors of constant (6): ±{1, 2, 3, 6}
Factors of leading (1): ±{1}
Candidates: 1, -1, 2, -2, 3, -3, 6, -6
Substitute x = 1 into the polynomial:
f(1) = (1)3 − 6(1)2 + 11(1) − 6 = 0
f(1) = 0 ✓ → (x − 1) is a factor
Perform synthetic division by (x − 1):
[1 | -6 | 11 | -6] ÷ (1) → [1 | -5 | 6]
Quotient: x2 − 5x + 6
Factor x^2 − 5x + 6:
= (x - 3)(x - 2)
= (x - 3)(x - 2)(x - 1)
(x - 3)(x - 2)(x - 1)
How to factor x^3-6x^2+11x-6
To factor x^3-6x^2+11x-6, identify the type of polynomial and apply the appropriate technique. The factored form is (x - 3)(x - 2)(x - 1).
This is a polynomial factoring problem — we break a polynomial into a product of simpler expressions. Common methods include GCF extraction, difference of squares, and trinomial factoring.
Frequently asked questions
What is the answer to x^3-6x^2+11x-6?
The answer is (x - 3)(x - 2)(x - 1).
What method is used?
identifying the polynomial type and applying the matching technique — GCF, trinomial factoring, difference of squares, or special cube formulas.