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Factor Out A Factor with A Negative Coefficient Calculator

Reviewed by Calculator Editorial Team

Factoring polynomials is a fundamental algebra skill. When you need to factor out a term with a negative coefficient, there's a specific method to follow. This calculator helps you perform this operation quickly and accurately.

Introduction

Factoring out a common term from a polynomial is a key algebraic technique. When the common term has a negative coefficient, you need to be careful about the signs when distributing the negative factor back into the expression.

For example, in the expression 3x - 6, you can factor out 3x. However, if you have -3x + 6, you factor out -3x. This calculator will guide you through the process step by step.

How to Use the Calculator

Using our calculator is simple:

  1. Enter the coefficients and variables for each term in your polynomial expression.
  2. Click the "Calculate" button to see the factored form.
  3. Review the step-by-step solution provided.
  4. Use the reset button to clear the calculator for a new problem.

The calculator handles up to four terms in the polynomial, which covers most common factoring problems.

The Method Explained

To factor out a term with a negative coefficient:

  1. Identify the greatest common factor (GCF) of all terms, including the negative sign.
  2. Factor the GCF out of each term.
  3. Write the GCF as the first factor, followed by the remaining expression in parentheses.

Formula: a(x + b) = ax + ab

When a is negative, the parentheses will contain terms with opposite signs.

For example, factoring -2x + 4:

  1. GCF is -2 (the negative coefficient).
  2. Factor out -2: -2(x) + -2(2).
  3. Combine: -2(x + 2).

Worked Examples

Example 1: Simple Negative Coefficient

Expression: -5x + 10

Solution:

  1. GCF is -5.
  2. Factor out -5: -5(x) + -5(2).
  3. Combine: -5(x + 2).

Final factored form: -5(x + 2)

Example 2: Multiple Variables

Expression: -3xy + 6x

Solution:

  1. GCF is -3x.
  2. Factor out -3x: -3x(y) + -3x(2).
  3. Combine: -3x(y + 2).

Final factored form: -3x(y + 2)

Example 3: Four Terms

Expression: -2a + 4b - 6c + 8d

Solution:

  1. GCF is -2.
  2. Factor out -2: -2(a) + -2(2b) + -2(3c) + -2(4d).
  3. Combine: -2(a + 2b + 3c + 4d).

Final factored form: -2(a + 2b + 3c + 4d)

FAQ

What if the polynomial has a constant term?

The constant term is treated like any other term when finding the GCF. For example, in -4x + 8, the GCF is -4, and the factored form is -4(x - 2).

Can I factor out a negative variable?

Yes, you can factor out a negative variable just like a negative coefficient. For example, -x + 2 becomes -(x - 2).

What if there's no common factor?

If there's no common factor, the polynomial is already in its simplest form. The calculator will indicate this.