SAT Math Practice Question #419 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #419

Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.

The polynomial \( P(x) = x^3 + ax^2 + bx + c \) has roots at \( x = 2 \) and \( x = -3 \). When \( P(x) \) is divided by \( (x - 1) \), the remainder is \( -12 \). What is the value of the constant \( c \)?
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Socratic AI Engine Step-by-Step Derivation
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Tactical Insight (Hint 1)

Identify the quadratic form: consider converting to vertex form \( y = a(x-h)^2 + k \) or using \( x = - rac{b}{2a} \) to find the line of symmetry.

Elimination Framework (Hint 2)

Desmos Tactical Shortcut: Plot the left-hand side and right-hand side as separate equations (e.g. \( y_1 = f(x) \) and \( y_2 = g(x) \)). Click the intersection points directly to read exact coordinates.

Masterclass Solution & Distractor Trap Analysis

This is a quintessential SAT 800-level polynomial question that fuses the Factor Theorem with the Remainder Theorem. Because \( x = 2 \) and \( x = -3 \) are roots of the cubic polynomial \( P(x) \), we know that \( (x - 2) \) and \( (x + 3...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

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