Polynomial Expressions and Factoring
Expanding and factoring are the same multiplication run in opposite directions, and a short set of patterns covers what the SAT asks: two binomials multiplied term by term, a squared sum keeping its doubled middle term, and a difference of squares splitting into conjugate factors. The algebra is short. The points leak in three moves: the difference of squares mishandled or a sum of squares factored as if it split, a sign or term slip inside an otherwise right method, and correct work with the wrong object handed back, a factor when a root was asked.
These patterns aren't really about whether you can multiply binomials. They're about whether the middle term survived the square, whether every sign came through the factor pair intact, and whether you handed back the factor or the root the item actually named.
The work
4 ways in · any order
Lesson
Polynomial Expressions and Factoring
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Expand products and squares without dropping the middle term, factor quadratics with the signs straight, and recognize the difference of squares next to the sum that does not factor. The lesson works the method and the three patterns that derail it, and it saves the trap for last: correct factoring with the wrong object handed back.
Diagnostic
10-item topic check
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Ten items across the three patterns: mishandling the difference of squares, a sign or term slip when expanding or factoring, and reporting a factor when a root was asked, or the reverse. Take it cold to surface the ones still catching you, or after the lesson to confirm they are gone.
Grid-in Check
Student-produced response
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About a quarter of SAT math answers are typed, not chosen, with no options to react to. These grid-in items diagnose by the value you enter, then route into the same drills the multiple-choice check feeds.
Targeted Practice
Drill a single pattern
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Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the pattern and you move on.