Difference between revisions of "2017 AMC 12B Problems/Problem 12"
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<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ \sqrt{2}+2\sqrt{3} \qquad \textbf{(D)}\ 2\sqrt{2}+\sqrt{6} \qquad \textbf{(E)}\ (1+\sqrt{3}) + (1+\sqrt{3})i</math> | <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ \sqrt{2}+2\sqrt{3} \qquad \textbf{(D)}\ 2\sqrt{2}+\sqrt{6} \qquad \textbf{(E)}\ (1+\sqrt{3}) + (1+\sqrt{3})i</math> | ||
− | ==Solution== | + | ==Solution 1== |
The root of any polynomial of the form <math>z^n = a</math> will have all <math>n</math> of it roots will have magnitude <math>\sqrt[n]{a}</math> and be the vertices of a regular <math>n</math>-gon in the complex plane (This concept is known as the Roots of Unity). For the equation <math>z^{12} = 64</math>, it is easy to see <math>\pm\sqrt{2}</math> and <math>\pm {i} \sqrt{2}</math> as roots. Graphing these in the complex plane, we have four vertices of a regular dodecagon. Since the roots must be equally spaced, besides <math>\sqrt{2}</math>, there are four more roots with positive real parts lying in the first and fourth quadrants. We also know that the angle between these roots is <math>30^{\circ}</math>. We only have to find the real parts of the roots lying in the first quadrant, because the imaginary parts would cancel out with those from the fourth quadrant. We have two <math>30-60-90</math> triangles (the triangles formed by connecting the origin to the roots, and dropping a perpendicular line from each root to the real-axis), both with hypotenuse <math>\sqrt{2}</math>. This means that one has base <math>\frac{\sqrt{2}}{2}</math> and the other has base <math>\frac{\sqrt{6}}{2}</math>. Adding these and multiplying by two, we get the sum of the four roots as <math>\sqrt{2} + \sqrt{6}</math>. However, we have to add in the original solution of <math>\sqrt{2}</math>, so the answer is <math>\boxed{\textbf{(D)}2\sqrt{2} + \sqrt{6}}</math>. | The root of any polynomial of the form <math>z^n = a</math> will have all <math>n</math> of it roots will have magnitude <math>\sqrt[n]{a}</math> and be the vertices of a regular <math>n</math>-gon in the complex plane (This concept is known as the Roots of Unity). For the equation <math>z^{12} = 64</math>, it is easy to see <math>\pm\sqrt{2}</math> and <math>\pm {i} \sqrt{2}</math> as roots. Graphing these in the complex plane, we have four vertices of a regular dodecagon. Since the roots must be equally spaced, besides <math>\sqrt{2}</math>, there are four more roots with positive real parts lying in the first and fourth quadrants. We also know that the angle between these roots is <math>30^{\circ}</math>. We only have to find the real parts of the roots lying in the first quadrant, because the imaginary parts would cancel out with those from the fourth quadrant. We have two <math>30-60-90</math> triangles (the triangles formed by connecting the origin to the roots, and dropping a perpendicular line from each root to the real-axis), both with hypotenuse <math>\sqrt{2}</math>. This means that one has base <math>\frac{\sqrt{2}}{2}</math> and the other has base <math>\frac{\sqrt{6}}{2}</math>. Adding these and multiplying by two, we get the sum of the four roots as <math>\sqrt{2} + \sqrt{6}</math>. However, we have to add in the original solution of <math>\sqrt{2}</math>, so the answer is <math>\boxed{\textbf{(D)}2\sqrt{2} + \sqrt{6}}</math>. | ||
− | Solution by | + | Solution by vedadehhc |
+ | |||
+ | ==Solution 2== | ||
+ | <math>z^{12}=64</math> has a factor of <math>\sqrt{2}</math>, so we need to remember to multiply our solution below, using the Roots of Unity. We notice that the sum of the complex parts of all these roots is <math>0</math>, because the points on the complex plane are symmetric. The roots with <math>re(z)>0</math> are <math>e^{0i}, e^{\frac{\pm\pi}{6}i},</math> and <math>e^{\frac{\pm\pi}{3}i}</math> by the Roots of Unity. Their real parts are <math>\cos(0), \pm\cos(\frac{\pi}{6}),</math> and <math>\pm\cos(\frac{\pi}{3})</math>. Their sum is <math>1+2(\frac{\sqrt{3}}{2}+\frac{1}{2})=1+\sqrt{3}+1=2+\sqrt{3}</math>. But, remember to multiply by <math>\sqrt{2}</math>. The answer is <math>\sqrt{2}(2+\sqrt{3})=\boxed{\textbf{(D)}\ 2\sqrt{2}+\sqrt{6}}</math>. | ||
+ | |||
+ | Solution by TheUltimate123 (Eric Shen) | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC12 box|year=2017|ab=B|num-b=11|num-a=13}} | ||
+ | {{MAA Notice}} |
Revision as of 22:54, 16 February 2017
Contents
Problem 12
What is the sum of the roots of that have a positive real part?
Solution 1
The root of any polynomial of the form will have all of it roots will have magnitude and be the vertices of a regular -gon in the complex plane (This concept is known as the Roots of Unity). For the equation , it is easy to see and as roots. Graphing these in the complex plane, we have four vertices of a regular dodecagon. Since the roots must be equally spaced, besides , there are four more roots with positive real parts lying in the first and fourth quadrants. We also know that the angle between these roots is . We only have to find the real parts of the roots lying in the first quadrant, because the imaginary parts would cancel out with those from the fourth quadrant. We have two triangles (the triangles formed by connecting the origin to the roots, and dropping a perpendicular line from each root to the real-axis), both with hypotenuse . This means that one has base and the other has base . Adding these and multiplying by two, we get the sum of the four roots as . However, we have to add in the original solution of , so the answer is .
Solution by vedadehhc
Solution 2
has a factor of , so we need to remember to multiply our solution below, using the Roots of Unity. We notice that the sum of the complex parts of all these roots is , because the points on the complex plane are symmetric. The roots with are and by the Roots of Unity. Their real parts are and . Their sum is . But, remember to multiply by . The answer is .
Solution by TheUltimate123 (Eric Shen)
See Also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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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