Factoring out the greatest common factor is a core algebra skill that helps turn a sum of terms into a product. It matters because many algebra problems become simpler once a common factor has been pulled out. This method is used to simplify expressions, solve equations, graph functions, and work with polynomials.
The main idea is to find the largest factor shared by every term and place it outside parentheses.
Factoring out the GCF is the reverse of the distributive property. If 6x + 9 becomes 3(2x + 3), then distributing the 3 gives back 6x + 9. To factor correctly, look for common number factors and common variable factors with the smallest exponent shared by all terms.
After the GCF is removed, each term inside the parentheses is what remains after division by the GCF.
Key Facts
- Factoring rewrites a sum or difference as a product.
- The GCF is the greatest factor shared by every term in an expression.
- Distributive property: a(b + c) = ab + ac.
- Factoring out the GCF reverses distributing: ab + ac = a(b + c).
- For variables, use the smallest exponent found in every term, such as GCF of x^5 and x^2 is x^2.
- Example: 12x^3 + 18x^2 = 6x^2(2x + 3).
Vocabulary
- Greatest Common Factor
- The greatest common factor, or GCF, is the largest factor that divides every term in an expression.
- Factor
- A factor is a number or expression that is multiplied by another factor to make a product.
- Term
- A term is a single number, variable, or product of numbers and variables in an expression.
- Coefficient
- A coefficient is the numerical factor multiplied by a variable in a term.
- Distributive Property
- The distributive property says that multiplying a factor by a sum gives the same result as multiplying each term separately.
Common Mistakes to Avoid
- Factoring out a number that is not common to every term is wrong because the factor outside parentheses must divide all terms exactly.
- Forgetting the variable part of the GCF is wrong because common variables should also be pulled out, using the smallest exponent shared by all terms.
- Dropping a term when it divides to 1 is wrong because a term like 5x divided by 5x leaves 1, so the 1 must appear inside the parentheses if needed.
- Not checking by distributing is wrong because distributing the outside factor should reproduce the original expression exactly.
Practice Questions
- 1 Factor out the GCF: 24x + 36.
- 2 Factor out the GCF: 15a^4b^2 - 25a^2b^5.
- 3 Explain why factoring 8x + 12 as 2(4x + 6) is correct but not fully factored.