Factoring Higher-Degree Polynomials Calculator

Published on: December 15, 2024

This Factoring Higher-Degree Polynomials Calculator helps you factor polynomials of degree 3 or more into simpler factors step by step. It may use methods such as finding a greatest common factor, grouping, or testing possible roots to break the polynomial into lower-degree parts. The final result is written as a product of irreducible factors where possible. It is a simple way to check answers, understand polynomial factoring, and practise basic algebra step by step.

Step-by-step method

  1. List possible rational roots using the constant term and leading coefficient if needed.
  2. Test a root by substitution and show the arithmetic.
  3. Use synthetic division to reduce the degree.
  4. 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: Start with the polynomial.

x^3 − 6x^2 + 11x − 6

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

In this problem: Constant term:

−6

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

In this problem: Factors of 6:

1, 2, 3, 6

Step 1d - 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 2z - 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.

In this problem: Synthetic division:

Bring down 1

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

In this problem: Synthetic division:

1 ⋅ 1 = 1

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

In this problem: Synthetic division:

−6 + 1 = −5

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

In this problem: Synthetic division:

1 ⋅ −5 = −5

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

In this problem: Synthetic division:

11 + ( −5 ) = 6

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

In this problem: Synthetic division:

1 ⋅ 6 = 6

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

In this problem: Synthetic division:

−6 + 6 = 0

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

In this problem: Quotient:

x^2 − 5x + 6

Step 4a - Factor what remains and multiply the factors together.

In this problem: List factor pairs of 6:

(1,6), (2,3), (3,2), (6,1)

Step 4b - Factor what remains and multiply the factors together.

In this problem: Check sums to get −5:

−2 + ( −3 ) = −5

Step 4c - Factor what remains and multiply the factors together.

In this problem: Write the factorization:

x^2 − 5x + 6 = (x − 2)(x − 3)

Final answer: (x − 1)(x − 2)(x − 3)

Example 2: x^3 + x^2 - 4x - 4

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

In this problem: Start with the polynomial.

x^3 + x^2 − 4x − 4

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

In this problem: Constant term:

−4

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

In this problem: Factors of 4:

1, 2, 4

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

In this problem: Possible roots:

−1, 1, −2, 2, −4, 4

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

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

f( −1 ) = −1^3 + −1^2 −4( −1 ) −4

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

In this problem: Evaluate powers:

−1 + 1 −4( −1 ) −4

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

In this problem: Evaluate multiplications:

−1 + 1 + 4 − 4

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

In this problem: Combine left-to-right:

−1 + 1 = 0

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

In this problem: Combine left-to-right:

0 + 4 = 4

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

In this problem: Combine left-to-right:

4 − 4 = 0

Step 2z - 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, 1, −4, −4 by root −1.

1, 1, −4, −4

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

In this problem: Synthetic division:

Bring down 1

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

In this problem: Synthetic division:

−1 ⋅ 1 = −1

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

In this problem: Synthetic division:

1 + ( −1 ) = 0

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

In this problem: Synthetic division:

−1 ⋅ 0 = 0

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

In this problem: Synthetic division:

−4 + 0 = −4

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

In this problem: Synthetic division:

−1 ⋅ −4 = 4

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

In this problem: Synthetic division:

−4 + 4 = 0

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

In this problem: Quotient:

x^2 − 4

Step 4a - Factor what remains and multiply the factors together.

In this problem: List factor pairs of 4:

(1,4), (2,2), (4,1)

Step 4b - Factor what remains and multiply the factors together.

In this problem: Check sums to get 0:

2 + ( −2 ) = 0

Step 4c - Factor what remains and multiply the factors together.

In this problem: Write the factorization:

x^2 − 4 = (x + 2)(x − 2)

Final answer: (x + 1)(x + 2)(x − 2)