SAT Math Practice Question #380 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #380

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

The polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) has roots at \( x = 1 \), \( x = -2 \), and \( x = 3 \). When \( P(x) \) is divided by \( x + 1 \), the remainder is \( -24 \). What is the value of the constant \( a \)?
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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 problem tests your mastery of the Remainder Theorem, polynomial roots, and Vieta's formulas. Step 1: Set up the polynomial. Since \( P(x) \) is a degree 4 polynomial (indicated by \( x^4 \)) with a leading coefficient of 1, it must hav...

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

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