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Test Citadel features an authentic Bluebook-calibrated countdown timer with visibility toggle, question navigation grid, choice eliminator cross-out tool, embedded Desmos graphing calculator, and official College Board formula sheets.

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Yes. Every visitor can complete 4 free diagnostic problems (2 Math with Desmos and 2 Reading & Writing) with full Socratic explanations before upgrading to Citadel Pro.

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Citadel Pro grants unlimited access to 2,600+ adaptive questions, multi-tier Socratic hint trees, Desmos graphing shortcuts, distractor trap breakdowns, and simulated full-length SAT and ACT exams.

SAT Math Difficulty: Hard (800) • Calibrated to 2026 Testing Standard

Digital SAT Math Practice Question #466

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

Let $ P(x) = x^4 + ax^3 + bx^2 + cx + d $, where $ a, b, c, $ and $ d $ are real constants. When the polynomial $ P(x) $ is divided by $ x^2 + 1 $, the remainder is $ 3x + 2 $. When $ P(x) $ is divided by $ x^2 - 2x + 2 $, the remainder is $ 5x - 4 $. What is the value of $ a + b + c + d $?

Select Your Answer:

Socratic AI Engine Step-by-Step Derivation

0ms Precomputed

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)

Elimination Shortcut: Check the extremes or test answer choices starting with C to binary search the correct magnitude.

Masterclass Solution & Distractor Trap Analysis

Alright, my mathematically gifted student, buckle up. This is a true 800-level beast that tests your mastery of the Polynomial Remainder Theorem, complex numbers, and system of equations. Let's conquer it step by step. **Step 1: Understand ...

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

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