2003 AMC 12B Problem 20 — Solution | LIVE by Po-Shen Loh

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2003 AMC 12B Problem 20

Attempt Problem 20 of the 2003 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 2003 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

20.

Part of the graph of $f(x) = ax^3 + bx^2 + cx + d$ is shown. What is $b?$

$-4$

$-2$

$0$

$2$

$4$

Answer: B

Concepts:polynomial substitution

Difficulty rating: 1580

Solution:

The graph passes through $(-1, 0),$ $(1, 0),$ and $(0, 2).$ So $f(0) = d = 2.$

Adding $\begin{aligned} &f(1) + f(-1) \\ &= (a + b + c + d) \\ &\quad {}+ (-a + b - c + d) \\ &= 2b + 2d = 0, \end{aligned}$ so $b = -d = -2.$

Thus, the correct answer is B.

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