Factoring: GCF and Grouping

Calculus Readiness · Axiom Academy

The first move in every factoring problem: find the greatest common factor of the terms and pull it out front. 1. Pull the Common Factor Out Front Look at 6x^2 + 9x . Break each term into its factors and the shared piece jumps out: both carry a 3 and both carry an x . That shared 3x is the GCF — slide it out front as a single multiplier and whatever is left stays inside the parentheses. The shared 3x is the greatest common factor Pulled out front — the leftovers stay inside 2. Find the GCF, Then Divide Each Term The GCF has two halves. Take the biggest number that divides every coefficient, and for each variable take the lowest power that appears in all terms. Then divide every term by that GCF to read off what is left. Watch it on 10x^4 - 15x^3 + 25x^2 . — the largest integer dividing every coefficient. x appears as x^4, x^3, x^2 . The lowest power, x^2 , is shared by all. A term divided by itself leaves 1 , not nothing — keep it inside. To divide every term you can only pull out as many x 's as the stingiest term owns. Here x^2 is the floor, so x^2 comes out and the residual 2x^2 - 3x + 5 has no common factor left. Every factoring should survive a check: distribute the GCF back across the parentheses and you must land on the original. Take the classic 12x^3 + 18x^2 + 6x , whose GCF is 6x . Multiply 6x into each inner term and the original reappears, term for term. Fan it out: , then , then — each product rebuilds one term of the original.

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