Factoring Out The GCF
Factor out the Greatest Common Factor (GCF) in Grade 9 algebra as the first step in any factoring problem: find the largest factor shared by all terms, then divide it out using the distributive property in reverse.
Key Concepts
Property To factor a polynomial completely, first identify and factor out the Greatest Common Factor (GCF) from all terms. The GCF is the largest monomial that divides into each term of the polynomial. Explanation Think of the GCF as the biggest shared ingredient in every term. Before you start solving the main trinomial puzzle, pull that GCF out to the front! This step simplifies the remaining expression, making it much smaller and easier to work with. It's like tidying up your desk before starting homework—it just makes everything clearer and simpler. Examples $x^5 + 6x^4 + 8x^3 = x^3(x^2 + 6x + 8) = x^3(x + 2)(x + 4)$ $5x^3 10x^2 120x = 5x(x^2 2x 24) = 5x(x + 4)(x 6)$.
Common Questions
What is the GCF and how do you find it for a polynomial?
The GCF is the largest factor that divides evenly into every term of the polynomial. Find the GCF of the coefficients, then identify the lowest power of each common variable. For 6x³ + 9x², GCF = 3x².
How do you factor out the GCF from a polynomial?
Divide every term by the GCF, then write the GCF outside the parentheses and the quotients inside. For 6x³ + 9x² with GCF = 3x²: factor as 3x²(2x + 3). Verify by redistributing.
Why should you always factor out the GCF first?
Removing the GCF simplifies the remaining polynomial, making further factoring (trinomial factoring, grouping, etc.) much easier. Working with smaller coefficients reduces the chance of arithmetic errors.