SAT Math Practice Question #507 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #507

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

The polynomial \( P \) is defined by \( P(x) = x^3 + ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The equation \( P(x) = 0 \) has roots at \( x = -3 \), \( x = 2 \), and \( x = 6 \). The polynomial \( Q \) is defined by \( Q(x) = P(x - k) \), where \( k \) is a constant. If \( Q(x) \) is evenly divisible by \( (x + 1) \), which of the following could be the value of \( k \)?
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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

To solve this, you need a masterful grasp of the Factor Theorem and function shifts. Step 1: Understand the roots of \( P(x) \). We are given that the roots of \( P(x) = 0 \) are \( -3 \), \( 2 \), and \( 6 \). This means that: \( P(-3) = 0...

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

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