Factoring Higher-Degree Polynomials Calculator

Published on: December 15, 2024
Final Answer: Free • Full Steps: Plus

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:

\(\text{Bring down } 1\)

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

In this problem: Synthetic division:

\(1 \cdot 1 = 1\)

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:

\(1 \cdot -5 = -5\)

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:

\(1 \cdot 6 = 6\)

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:

\(\text{Bring down } 1\)

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

In this problem: Synthetic division:

\(-1 \cdot 1 = -1\)

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:

\(-1 \cdot 0 = 0\)

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:

\(-1 \cdot -4 = 4\)

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:

\((x + 1)(x + 2)(x - 2)\)
See Example 1 Hide Example 1

Problem

\(x^3 - 6x^2 + 11x - 6\)

Approach

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.

Step 1a

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

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

In this problem: Constant term:

\(-6\)

Step 1c

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

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

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

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

In this problem: Evaluate powers:

\(1 -6( 1 ) + 11( 1 ) -6\)

Step 2c

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

In this problem: Evaluate multiplications:

\(1 - 6 + 11 - 6\)

Step 2d

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

In this problem: Combine left-to-right:

\(1 - 6 = -5\)

Step 2e

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

In this problem: Combine left-to-right:

\(-5 + 11 = 6\)

Step 2f

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

In this problem: Combine left-to-right:

\(6 - 6 = 0\)

Step 2z

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

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

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

In this problem: Synthetic division:

\(\text{Bring down } 1\)

Step 3b

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

In this problem: Synthetic division:

\(1 \cdot 1 = 1\)

Step 3c

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

In this problem: Synthetic division:

\(-6 + 1 = -5\)

Step 3d

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

In this problem: Synthetic division:

\(1 \cdot -5 = -5\)

Step 3e

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

In this problem: Synthetic division:

\(11 + ( -5 ) = 6\)

Step 3f

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

In this problem: Synthetic division:

\(1 \cdot 6 = 6\)

Step 3g

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

In this problem: Synthetic division:

\(-6 + 6 = 0\)

Step 3z

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

In this problem: Quotient:

\(x^2 - 5x + 6\)

Step 4a

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

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

In this problem: Check sums to get −5:

\(-2 + ( -3 ) = -5\)

Step 4c

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

\(x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3)\)
See Example 2 Hide Example 2

Problem

\(x^3 + x^2 - 4x - 4\)

Approach

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.

Step 1a

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

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

In this problem: Constant term:

\(-4\)

Step 1c

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

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

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

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

In this problem: Evaluate powers:

\(-1 + 1 -4( -1 ) -4\)

Step 2c

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

In this problem: Evaluate multiplications:

\(-1 + 1 + 4 - 4\)

Step 2d

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

In this problem: Combine left-to-right:

\(-1 + 1 = 0\)

Step 2e

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

In this problem: Combine left-to-right:

\(0 + 4 = 4\)

Step 2f

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

In this problem: Combine left-to-right:

\(4 - 4 = 0\)

Step 2z

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

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

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

In this problem: Synthetic division:

\(\text{Bring down } 1\)

Step 3b

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

In this problem: Synthetic division:

\(-1 \cdot 1 = -1\)

Step 3c

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

In this problem: Synthetic division:

\(1 + ( -1 ) = 0\)

Step 3d

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

In this problem: Synthetic division:

\(-1 \cdot 0 = 0\)

Step 3e

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

In this problem: Synthetic division:

\(-4 + 0 = -4\)

Step 3f

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

In this problem: Synthetic division:

\(-1 \cdot -4 = 4\)

Step 3g

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

In this problem: Synthetic division:

\(-4 + 4 = 0\)

Step 3z

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

In this problem: Quotient:

\(x^2 - 4\)

Step 4a

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

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

In this problem: Check sums to get 0:

\(2 + ( -2 ) = 0\)

Step 4c

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

\(x^3 + x^2 - 4x - 4 = (x + 1)(x + 2)(x - 2)\)