Difference between revisions of "2021 AMC 12A Problems/Problem 22"

Revision as of 14:00, 12 February 2021

Problem

Suppose that the roots of the polynomial $P(x)=x^3+ax^2+bx+c$ are $\cos \frac{2\pi}7,\cos \frac{4\pi}7,$ and $\cos \frac{6\pi}7$ , where angles are in radians. What is $abc$ ?

$\textbf{(A) }-\frac{3}{49} \qquad \textbf{(B) }-\frac{1}{28} \qquad \textbf{(C) }\frac{^3\sqrt7}{64} \qquad \textbf{(D) }\frac{1}{32}\qquad \textbf{(E) }\frac{1}{28}$

Solution 1

Part 1: solving for c

Notice that $\cos \frac{6\pi}7 = \cos \frac{8\pi}7$

$c$ is the negation of the product of roots by Vieta's formulas

$c = -\cos \frac{2\pi}7 \cos \frac{4\pi}7 \cos \frac{8\pi}7$

Multiply by $8 \sin{\frac{2\pi}{7}}$

$c 8 \sin{2\pi}7 = -8 \sin{\frac{2\pi}{7}} \cos \frac{2\pi}7 \cos \frac{4\pi}7 \cos \frac{8\pi}7$

Then use sine addition formula backwards:

$2 \sin \frac{2\pi}7 \cos \frac{2\pi}7 = \sin \frac{4\pi}7$

$c \cdot 8 \sin{\frac{2\pi}{7}} = -4 \sin \frac{4\pi}7 \cos \frac{4\pi}7 \cos \frac{8\pi}7$

$c \cdot 8 \sin{\frac{2\pi}{7}} = -2 \sin \frac{8\pi}7 \cos \frac{8\pi}7$

$c \cdot 8 \sin{\frac{2\pi}{7}} = -\sin \frac{16\pi}7$

$c \cdot 8 \sin{\frac{2\pi}{7}} = -\sin \frac{2\pi}7$

$c = -\frac{1}8$

Part 2: starting to solve for b

$b$ is the sum of roots two at a time by Vieta's

$b = \cos \frac{2\pi}7 \cos \frac{4\pi}7 + \cos \frac{2\pi}7 \cos \frac{6\pi}7 + \cos \frac{4\pi}7 \cos \frac{6\pi}7$

We know that $\cos \alpha \cos \beta = \frac{ \cos \left(\alpha + \beta\right) + \cos \left(\alpha - \beta\right) }{2}$

By plugging all the parts in we get:

$\frac{\cos \frac{6\pi}7 + \cos \frac{2\pi}7}2 + \frac{\cos \frac{4\pi}7 + \cos \frac{4\pi}7}2 + \frac{\cos \frac{6\pi}7 + \cos \frac{2\pi}7}2$

Which ends up being:

$\cos \frac{2\pi}7 + \cos \frac{4\pi}7 + \cos \frac{6\pi}7$

Which is shown in the next part to equal $-\frac{1}2$ , so $b = -\frac{1}2$

Part 3: solving for a and b as the sum of roots

$a$ is the negation of the sum of roots

$a = - \left( \cos \frac{2\pi}7 + \cos \frac{4\pi}7 + \cos \frac{6\pi}7 \right)$

The real values of the 7th roots of unity are: $1, \cos \frac{2\pi}7, \cos \frac{4\pi}7, \cos \frac{6\pi}7, \cos \frac{8\pi}7, \cos \frac{10\pi}7, \cos \frac{12\pi}7$ and they sum to $0$ .

If we subtract 1, and condense identical terms, we get:

$2\cos \frac{2\pi}7 + 2\cos \frac{4\pi}7 + 2\cos \frac{6\pi}7 = -1$

Therefore, we have $a = -\left(-\frac{1}2\right) = \frac{1}2$

Finally multiply $abc = \frac{1}2 * - \frac{1}2 * -\frac{1}8 = \frac{1}{32}$ or $\boxed{D) \frac{1}{32}}$ .

~Tucker

Solution 2 (Approximation)

Letting the roots be $p$ , $q$ , and $r$ , Vietas gives $p + q + r = a$ $pq + qr + pq = -b$ $pqr = c$ We use the Taylor series for $\cos x$ , $\cos x = \sum_{k = 0}^{\infty} (-1)^k \frac{x^{2k}}{(2k)!}$ to approximate the roots. Taking the sum up to $k = 3$ yields a close approximation, so we have $\cos\left(\frac{2\pi}{7}\right) \simeq 1-\frac{\left(\frac{2\pi}{7}\right)^{2}}{2}+\frac{\left(\frac{2\pi}{7}\right)^{4}}{24}-\frac{\left(\frac{2\pi}{7}\right)^{6}}{720} \simeq 0.623$ $\cos\left(\frac{4\pi}{7}\right) \simeq 1-\frac{\left(\frac{4\pi}{7}\right)^{2}}{2}+\frac{\left(\frac{4\pi}{7}\right)^{4}}{24}-\frac{\left(\frac{4\pi}{7}\right)^{6}}{720} \simeq -0.225$ $\cos\left(\frac{6\pi}{7}\right) \simeq 1-\frac{\left(\frac{6\pi}{7}\right)^{2}}{2}+\frac{\left(\frac{6\pi}{7}\right)^{4}}{24}-\frac{\left(\frac{6\pi}{7}\right)^{6}}{720} \simeq -0.964.$ Note that these approximations get worse as $x$ gets larger, but they will be fine for the purposes of this problem. We then have $p + q + r = a \simeq -0.56$ $pq + qr + pr = -b \simeq -0.524$ $pqr = c \simeq 0.135$ We further approximate these values to $a \simeq -0.5$ , $b \simeq 0.5$ , and $c \simeq 0.125$ (mostly as this is an AMC problem and will likely use nice fractions). Thus, we have $abc \simeq \boxed{\textbf{(D) } \frac{1}{32}}$ . ~ciceronii

Remark: In order to be more confident in your answer, you can go a few terms further in the Taylor series.

Video Solution by OmegaLearn (Euler's Identity + Vieta's )

https://youtu.be/Im_WTIK0tss

~ pi_is_3.14

@@ Line 82: / Line 82: @@
 <cmath>\cos\left(\frac{6\pi}{7}\right) \simeq 1-\frac{\left(\frac{6\pi}{7}\right)^{2}}{2}+\frac{\left(\frac{6\pi}{7}\right)^{4}}{24}-\frac{\left(\frac{6\pi}{7}\right)^{6}}{720} \simeq -0.964.</cmath>
 Note that these approximations get worse as <math>x</math> gets larger, but they will be fine for the purposes of this problem. We then have
-<cmath>p + q + r &= a \simeq -0.56</cmath>
+<cmath>p + q + r = a \simeq -0.56</cmath>
-<cmath>pq + qr + pr &= -b \simeq -0.524</cmath>
+<cmath>pq + qr + pr = -b \simeq -0.524</cmath>
 <cmath>pqr = c \simeq 0.135</cmath>
 We further approximate these values to <math>a \simeq -0.5</math>, <math>b \simeq 0.5</math>, and <math>c \simeq 0.125</math> (mostly as this is an AMC problem and will likely use nice fractions). Thus, we have <math>abc \simeq \boxed{\textbf{(D) } \frac{1}{32}}</math>. ~ciceronii
-\textbf{Remark}: In order to be more confident in your answer, you can go a few terms further in the Taylor series.
+Remark: In order to be more confident in your answer, you can go a few terms further in the Taylor series.
 == Video Solution by OmegaLearn (Euler's Identity + Vieta's ) ==

2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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