Factor Out The Negative of The Greatest Common Factor Calculator

Factoring out the negative of the greatest common factor (GCF) is a fundamental algebraic operation that simplifies expressions by identifying and removing the largest negative factor common to all terms. This process is essential in algebra, calculus, and engineering for simplifying equations, solving problems, and preparing expressions for further operations.

What is the Greatest Common Factor (GCF)?

The greatest common factor (GCF) of a set of numbers or terms is the largest factor that divides each of them without leaving a remainder. For example, the GCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18.

When working with polynomials, the GCF is the largest polynomial that divides each term of the polynomial. For example, in the expression 6x² + 9x, the GCF is 3x because 3x divides both terms.

Note: The GCF is always a positive number or polynomial, even if the original terms contain negative numbers or coefficients.

How to Factor Out the Negative GCF

Factoring out the negative GCF involves these steps:

Identify the GCF of all terms in the expression.
If the GCF is negative, factor it out as a negative number.
Divide each term by the GCF and write the result inside parentheses.
Multiply the GCF by the expression in parentheses to verify the result.

Formula: If the GCF is -a, then the expression can be written as -a × (expression with positive coefficients).

For example, consider the expression -6x² + 9x. The GCF is -3x. Factoring out -3x gives:

-3x × (2x - 3)

This is equivalent to the original expression because -3x × 2x = -6x² and -3x × -3 = 9x.

Worked Examples

Example 1: Simple Polynomial

Expression: -8x² + 12x

GCF of 8 and 12 is 4. Since the expression is negative, the GCF is -4.
Divide each term by -4: (-8x²)/(-4) = 2x², (12x)/(-4) = -3x.
Factor out -4: -4 × (2x² - 3x).

Example 2: Mixed Terms

Expression: -15y³ + 20y² - 5y

GCF of 15, 20, and 5 is 5. Since the expression is negative, the GCF is -5.
Divide each term by -5: (-15y³)/(-5) = 3y³, (20y²)/(-5) = -4y², (-5y)/(-5) = y.
Factor out -5: -5 × (3y³ - 4y² + y).

Comparison of Original and Factored Forms
Original Expression	Factored Form
-8x² + 12x	-4 × (2x² - 3x)
-15y³ + 20y² - 5y	-5 × (3y³ - 4y² + y)

FAQ

What happens if the GCF is negative?

If the GCF is negative, you factor it out as a negative number. This ensures the coefficients inside the parentheses are positive, making the expression easier to work with.

Can I factor out the GCF if it's not negative?

Yes, you can still factor out the GCF even if it's positive. The process is the same, but the result will have a positive GCF.

What if the expression has only one term?

If the expression has only one term, the GCF is the term itself, and factoring out the GCF doesn't change the expression.