2002 AMC 12B Problem 2

Attempt Problem 2 of the 2002 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12B solutions, or check the answer key.

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2.

What is the value of (3x−2)(4x+1)−(3x−2)4x+1 \begin{aligned} &(3x-2)(4x+1) \\ &\quad {}-(3x-2)4x+1 \end{aligned} when x=4?x=4?
00
11
1010
1111
1212
Answer: D
Concepts:algebraic manipulationfactoring
Small hint:
Group the two terms containing 3x−23x-2 using the distributive law
Big hint:
(3x−2)(4x+1)−(3x−2)4x(3x-2)(4x+1)-(3x-2)4x =(3x−2)(4x+1−4x)=(3x-2)(4x+1-4x)
Solution:
Factor 3x−23x-2 out of the first two terms: (3x−2)(4x+1)−(3x−2)4x+1=(3x−2)(4x+1−4x)+1=3x−1. \begin{gathered} (3x-2)(4x+1) \\ {}-(3x-2)4x+1 \\ {}=(3x-2)(4x+1-4x)+1 \\ {}=3x-1. \end{gathered} At x=4x=4 this equals 3⋅4−1=11.3\cdot4-1=11.
Thus, the correct answer is D.
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