SAT Math Practice Question #706 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #706

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

The graph of the degree-4 polynomial \( f(x) = x^4 + bx^3 + cx^2 + dx + e \) intersects the graph of the linear function \( g(x) = mx + k \) at exactly three points: \( (-2, -5) \), \( (1, 4) \), and \( (3, 10) \). If the graph of \( f(x) \) is tangent to the graph of \( g(x) \) at \( x = 3 \), what is the value of \( e \)?
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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 is a quintessential 800-level polynomial problem that combines systems of equations, coordinate geometry, and the Remainder/Factor Theorems. Let's conquer it step-by-step. Step 1: Find the equation of the line \( g(x) \). We are given ...

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

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