Difference between revisions of "2003 AMC 12B Problems/Problem 20"

(Solution 3)
(Solution 4)
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implies by <math>x^2</math> coefficient, <math>b + 2 = 0, b = -2 \Rightarrow \mathrm{(B)}</math>.
 
implies by <math>x^2</math> coefficient, <math>b + 2 = 0, b = -2 \Rightarrow \mathrm{(B)}</math>.
  
==Solution 4==
+
===Solution 4===
 
The roots of this equation are <math>-1, 1, \text{ and } x</math>, letting <math>x</math> be the root not shown in the graph. By Vieta, we know that <math>-1+1+x=x=-\frac{b}{a}</math> and <math>-1\cdot 1\cdot x=-x=-\frac{d}{a}</math>. Therefore, <math>x=\frac{d}{a}</math>. Setting the two equations for <math>x</math> equal to each other, <math>\frac{d}{a}=-\frac{b}{a}</math>. We know that the y-intercept of the polynomial is <math>d</math>, so <math>d=2</math>. Plugging in for <math>d</math>, <math>\frac{2}{a}=-\frac{b}{a}</math>.
 
The roots of this equation are <math>-1, 1, \text{ and } x</math>, letting <math>x</math> be the root not shown in the graph. By Vieta, we know that <math>-1+1+x=x=-\frac{b}{a}</math> and <math>-1\cdot 1\cdot x=-x=-\frac{d}{a}</math>. Therefore, <math>x=\frac{d}{a}</math>. Setting the two equations for <math>x</math> equal to each other, <math>\frac{d}{a}=-\frac{b}{a}</math>. We know that the y-intercept of the polynomial is <math>d</math>, so <math>d=2</math>. Plugging in for <math>d</math>, <math>\frac{2}{a}=-\frac{b}{a}</math>.
  

Revision as of 19:11, 7 July 2020

Problem

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

2003 12B AMC-20.png

$\mathrm{(A)}\ -4 \qquad\mathrm{(B)}\ -2 \qquad\mathrm{(C)}\ 0 \qquad\mathrm{(D)}\ 2 \qquad\mathrm{(E)}\ 4$

Solution

Solution 1

Since \begin{align*} -f(-1) = a - b + c - d = 0 = f(1) = a + b + c + d \end{align*}

It follows that $b + d = 0$. Also, $d = f(0) = 2$, so $b = -2 \Rightarrow \mathrm{(B)}$.

Solution 2

Two of the roots of $f(x) = 0$ are $\pm 1$, and we let the third one be $n$. Then \[a(x-1)(x+1)(x-n) = ax^3-anx^2-ax+an = ax^3 + bx^2 + cx + d = 0\] Notice that $f(0) = d = an = 2$, so $b = -an = -2 \Rightarrow \mathrm{(B)}$.

Solution 3

Notice that if $g(x) = 2 - 2x^2$, then $f - g$ vanishes at $x = -1, 0, 1$ and so \[f(x) - g(x) = ax(x-1)(x+1) = ax^3 - ax\] implies by $x^2$ coefficient, $b + 2 = 0, b = -2 \Rightarrow \mathrm{(B)}$.

Solution 4

The roots of this equation are $-1, 1, \text{ and } x$, letting $x$ be the root not shown in the graph. By Vieta, we know that $-1+1+x=x=-\frac{b}{a}$ and $-1\cdot 1\cdot x=-x=-\frac{d}{a}$. Therefore, $x=\frac{d}{a}$. Setting the two equations for $x$ equal to each other, $\frac{d}{a}=-\frac{b}{a}$. We know that the y-intercept of the polynomial is $d$, so $d=2$. Plugging in for $d$, $\frac{2}{a}=-\frac{b}{a}$.

Therefore, $b=-2 \Rightarrow \boxed{B}$

See also

2003 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 19
Followed by
Problem 21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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