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Factoring out the greatest common factor (GCF) is the reverse of the distributive property: instead of expanding a(b + c) into ab + ac, you start with ab + ac and write it as a(b + c). To factor 6x³ + 9x², find the GCF of the coefficients (3) and the lowest power of x (x²), giving 3x²(2x + 3). This is always the first step in any factoring problem — before trying other methods, always check for a GCF. Factoring is essential for solving polynomial equations, simplifying rational expressions, and finding roots.
Practice finding the GCF of monomials first (both numerical and variable parts). Then factor it out of two-term, three-term, and four-term polynomials. Verify by redistributing. Emphasize that factoring out the GCF should always be the first step, even when other factoring techniques will follow. Include negative leading coefficients (factor out −1 when helpful).
Factoring out the GCF is the distributive property run in reverse. You already know how to expand: 3x²(2x + 3) gives you 6x³ + 9x² by multiplying the outside term into each term inside the parentheses. Factoring out the GCF asks the opposite question — given 6x³ + 9x², can you find what was "outside" and what was "inside"? The answer is found by identifying the largest factor that all terms share.
Finding the GCF has two parts: the numerical coefficient and the variable part. For 6x³ + 9x², the coefficients are 6 and 9; the greatest common factor of those numbers is 3. The variable parts are x³ and x²; the GCF uses the lowest exponent, which is x². Putting them together: the GCF is 3x². Now divide each term by 3x² to find what goes inside the parentheses: 6x³ ÷ 3x² = 2x, and 9x² ÷ 3x² = 3. So the factored form is 3x²(2x + 3). You can always verify by redistributing — expand 3x²(2x + 3) and confirm you get back the original expression.
The most common error is factoring out something smaller than the greatest common factor. If you only pull out 3 from 6x³ + 9x², you get 3(2x³ + 3x²), which is technically factored but not completely — there is still an x² hiding inside that could come out. The word "greatest" matters: you want the largest possible factor, not just any common factor. Similarly, don't forget the variable part: from 6x² + 9x, the GCF is 3x (not just 3), giving 3x(2x + 3).
Factoring out the GCF is always the first step in any factoring problem, before you try other techniques like factoring trinomials or difference of squares. Even when a polynomial needs several rounds of factoring, clearing the GCF first makes every subsequent step simpler. Think of it as tidying up before rearranging furniture — it makes everything else easier to see and handle. The factored form is also useful for solving equations: if you need to solve 6x³ + 9x² = 0, factoring to 3x²(2x + 3) = 0 immediately reveals the solutions x = 0 (from the 3x² factor) and x = −3/2 (from the 2x + 3 factor).