Factoring Higher-Degree Polynomials Calculator

Published on: December 15, 2024

Enter a polynomial (e.g., x^3 - 6x^2 + 11x - 6) to factor it into irreducible factors.

List possible rational roots using the constant term (and leading coefficient if needed).
Test a root by substitution and show the arithmetic.
Use synthetic division to reduce the degree.
Factor what remains and multiply the factors together.

Example 1: x^3 - 6x^2 + 11x - 6

Step 1a - List possible rational roots using the constant term (and leading coefficient if needed).

In this problem: Constant term =

−6

Step 1b - List possible rational roots using the constant term (and leading coefficient if needed).

In this problem: Factors of 6:

1, 2, 3, 6

Step 1c - List possible rational roots using the constant term (and leading coefficient if needed).

In this problem: Possible roots:

−1, 1, −2, 2, −3, 3, −6, 6

Step 2a - Test a root by substitution and show the arithmetic.

In this problem: Test x = 1. Substitute x = 1:

f( 1 ) = 1^3 −6(1^2) + 11( 1 ) −6

Step 2b - Test a root by substitution and show the arithmetic.

In this problem: Evaluate powers:

1 −6( 1 ) + 11( 1 ) −6

Step 2c - Test a root by substitution and show the arithmetic.

In this problem: Evaluate multiplications:

1 −6 + 11 −6

Step 2d - Test a root by substitution and show the arithmetic.

In this problem: Combine left-to-right:

1 − 6 = −5

Step 2e - Test a root by substitution and show the arithmetic.

In this problem: Combine left-to-right:

−5 + 11 = 6

Step 2f - Test a root by substitution and show the arithmetic.

In this problem: Combine left-to-right:

6 − 6 = 0

Step 2f - Test a root by substitution and show the arithmetic.

In this problem: Since f( 1 ) = 0, (x − 1) is a factor.

f( 1 ) = 0

Step 3 - Use synthetic division to reduce the degree.

In this problem: Divide coefficients 1, −6, 11, −6 by root 1.

1, −6, 11, −6

Step 3a - Use synthetic division to reduce the degree.