ACT Preparation / Unit 8 ACT 0714 worked example, 60 to 90 seconds
Factoring Completely
The idea
To factor an expression completely, work in four steps. First, take out the greatest common factor. Second, count the terms of what remains: two terms suggest a difference of squares or cubes, three suggest a trinomial pattern or grouping, and four suggest grouping in pairs. Third, factor by the matching method. Fourth, inspect each new factor and keep factoring until none can be factored further.
In practice
Worked example: factor 3x⁴ − 48. Step one: 3(x⁴ − 16). Step two: two terms, both squares: x⁴ = (x²)² and 16 = 4². Step three: 3(x² + 4)(x² − 4). Step four: x² − 4 is again a difference of squares, giving 3(x² + 4)(x + 2)(x − 2). The factor x² + 4 is a sum of squares and stops there.
Where people slip
Stopping one step early is the error to check for: 3(x² + 4)(x² − 4) is factored, but not completely.
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
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