SAT Polynomials & Factoring Strategies

Polynomial questions test factoring techniques, the remainder theorem, and polynomial behavior. You'll see questions about finding roots, using the factor theorem (if x = a is a root, then (x−a) is a factor), and understanding what happens to a polynomial's output at specific inputs.

The SAT rarely asks for full polynomial long division — more often it asks whether a given value is a root, or what the remainder is when dividing by a linear factor. Recognizing the remainder theorem shortcut is often the difference between a 20-second solve and a 3-minute struggle.

Recognize the core factoring patterns: difference of squares (a²−b² = (a+b)(a−b)) and perfect square trinomials (a²+2ab+b² = (a+b)²). These appear constantly and should be instant.

The remainder theorem: the remainder when p(x) is divided by (x−a) equals p(a). Just substitute — no division needed. If asked "what is the remainder when p(x) is divided by (x−3)?", compute p(3).

Factor out the greatest common factor first, then apply other techniques. Missing the GCF creates unnecessary complexity.

Misapplying the remainder theorem by plugging in the wrong sign. For divisor (x−a), substitute x = +a, not x = −a. The sign matches what makes the divisor equal zero.

Forgetting to factor out the GCF before applying other techniques — this leaves a more complex expression than necessary and often makes factoring look impossible.

Assuming a degree-n polynomial always has n distinct real roots. Some roots may be complex, repeated, or irrational.

Drill difference of squares and perfect square trinomials until recognition is instant — these patterns appear in both explicit factoring and more subtle disguised forms on the SAT.

Practice the remainder theorem with several examples before test day: p(3), p(−2), p(0). The substitution is simple but students often revert to long division under pressure. ScoreBooster's Advanced Math sessions mix polynomial questions with quadratic questions to build broad algebraic recognition.