SAT Math Practice Question #617 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #617

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

The function \( f \) is defined by \( f(x) = \frac{x^3 + ax^2 + bx + c}{x^2 - 4x + 3} \), where \( a, b \), and \( c \) are constants. The graph of \( y = f(x) \) in the \(xy\)-plane has a slant (oblique) asymptote of \( y = x + 7 \) and a vertical asymptote ONLY at \( x = 1 \). If \( f(0) = 5 \), what is the value of \( a + b + c \)?
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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)

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 rational functions, asymptotes, and removable discontinuities (holes) at an absolute elite level. Let's conquer it step by step. Step 1: Analyze the Vertical Asymptote. The denominator of \( f(x) \) factor...

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

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