Difference between revisions of "2019 AMC 12A Problems/Problem 17"
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==Solution 1== | ==Solution 1== | ||
− | Applying Newton Sums we get the answer as 10 | + | Applying Newton's Sums (see [https://artofproblemsolving.com/wiki/index.php/Newton%27s_Sums this link]), we get the answer as <math>10</math>. |
==Solution 2== | ==Solution 2== | ||
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Therefore, <math>a = 5, b = -8</math>, and <math>c = 13</math> by matching coefficients. | Therefore, <math>a = 5, b = -8</math>, and <math>c = 13</math> by matching coefficients. | ||
− | <math>5 - 8 + 13 = \boxed{\textbf{(D)}10}</math> | + | <math>5 - 8 + 13 = \boxed{\textbf{(D)}10}</math>. |
==See Also== | ==See Also== |
Revision as of 19:22, 10 February 2019
Contents
[hide]Problem
Let denote the sum of the th powers of the roots of the polynomial . In particular, , , and . Let , , and be real numbers such that for , , What is ?
Solution 1
Applying Newton's Sums (see this link), we get the answer as .
Solution 2
Let , and be the roots of the polynomial. Then,
Adding these three equations, we get
can be written as .
We are given that is satisfied for , , , meaning it must be satisfied when , giving us .
Therefore, , and by matching coefficients.
.
See Also
2019 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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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