Polynomial Factoring Calculator

Step-by-step solution

1. 1

   Identify the polynomial

   The coefficients describe the polynomial x^2 - 5x + 6 of degree 2.

   $$P(x) = x^{2} - 5x + 6$$
2. 2

   Candidate rational roots

   By the rational root theorem, every rational root has the form p/q with p dividing 6 and q dividing 1.

   $$\frac{p}{q}, \quad p \mid 6, \quad q \mid 1$$

   • p ∣ 6: 1, 2, 3, 6
   • q ∣ 1: 1
   • ±p/q: ±1, ±2, ±3, ±6
3. 3

   Root found

   Substituting x = 2 into x^2 - 5x + 6 gives 0, so 2 is a root.

   $$P\left(2\right) = 0$$

   • b₂ = 1
   • b₁ = 1·(2) + (−5) = −3
   • b₀ = −3·(2) + (6) = 0
4. 4

   Synthetic division

   Divide by the linear factor of the root 2: synthetic division leaves the quotient x - 3.

   $$x^{2} - 5x + 6 = \left(x - 2\right) \cdot \left(x - 3\right)$$
5. 5

   Root found

   Substituting x = 3 into x - 3 gives 0, so 3 is a root.

   $$P\left(3\right) = 0$$

   • b₁ = 1
   • b₀ = 1·(3) + (−3) = 0
6. 6

   Synthetic division

   Divide by the linear factor of the root 3: synthetic division leaves the quotient 1.

   $$x - 3 = \left(x - 3\right) \cdot 1$$
7. 7

   Factored form

   The complete factorization of the polynomial over the rational numbers.

   $$P(x) = (x - 2)(x - 3)$$
8. 8

   Verify by expansion

   Expanding the product gives x^2 - 5x + 6, exactly the original polynomial x^2 - 5x + 6.

   $$(x - 2)(x - 3) = x^{2} - 5x + 6$$

   • (x - 2) · (x - 3) = x^2 - 5x + 6