SAT Math Practice Question #589 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #589

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 \( P(x) \) is divided by \( x - 2 \), the remainder is 5. When \( P(x) \) is divided by \( x + 1 \), the remainder is 5. If the polynomial \( P(x) - 5 \) is perfectly divisible by \( (x - 3)^2 \), what is the value of \( a + b + c + d \)?
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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

This question requires a masterful understanding of the Polynomial Remainder Theorem and algebraic structure. Trying to build a massive system of equations to solve for a, b, c, and d individually would take an hour. Let's use logic instead...

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

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