Difference between revisions of "2022 AMC 10B Problems/Problem 21"
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is <math>2x+1</math>. There is a unique polynomial of least degree with these two properties. What is the sum of | is <math>2x+1</math>. There is a unique polynomial of least degree with these two properties. What is the sum of | ||
the squares of the coefficients of that polynomial? | the squares of the coefficients of that polynomial? | ||
+ | |||
+ | ==Solution 1 (Experimentation)== | ||
+ | |||
+ | Given that all the answer choices and coefficients are integers, we hope that <math>P(x)</math> has positive integer coefficients. | ||
+ | |||
+ | Throughout this solution, we will express all polynomials in base <math>x</math>. E.g. <math>x^2 + x + 1 = 111_{x}</math>. | ||
+ | |||
+ | We are given: | ||
+ | |||
+ | <cmath>111a + 12 = 101b + 21 = P(x)</cmath>. | ||
+ | |||
+ | We add <math>111</math> and <math>101</math> to each side and balance respectively: | ||
+ | |||
+ | <cmath>111(a - 1) + 123 = 101(b - 1) + 122 = P(x)</cmath> | ||
+ | |||
+ | We make the units digits equal: | ||
+ | |||
+ | <cmath>111(a - 1) + 123 = 101(b - 2) + 223 = P(x)</cmath> | ||
+ | |||
+ | We now notice that: | ||
+ | |||
+ | <cmath>111(a - 11) + 1233 = 101(b - 12) + 1233 = P(x)</cmath>. | ||
+ | |||
+ | Therefore <math>a = 11_{x} = x + 1</math>, <math>b = 12_{x} = x + 2</math>, and <math>P(x) = 1233_{x} = x^3 + 2x^2 + 3x + 3</math>. <math>3</math> is the minimal degree of <math>P(x)</math> since there is no way to influence the <math>x</math>‘s digit in <math>101b + 21</math> when <math>b</math> is an integer. The desired sum is <math>1^2 + 2^2 + 3^2 + 3^2 = \boxed{\textbf{(E) 23}}</math> | ||
+ | |||
+ | P.S. The 4 computational steps can be deduced through quick experimentation. | ||
+ | |||
+ | ~ numerophile | ||
==Video Solutions== | ==Video Solutions== |
Revision as of 21:04, 18 November 2022
Problem
Let be a polynomial with rational coefficients such that when
is divided by the polynomial
, the remainder is
, and when
is divided by the polynomial
, the remainder
is
. There is a unique polynomial of least degree with these two properties. What is the sum of
the squares of the coefficients of that polynomial?
Solution 1 (Experimentation)
Given that all the answer choices and coefficients are integers, we hope that has positive integer coefficients.
Throughout this solution, we will express all polynomials in base . E.g.
.
We are given:
.
We add and
to each side and balance respectively:
We make the units digits equal:
We now notice that:
.
Therefore ,
, and
.
is the minimal degree of
since there is no way to influence the
‘s digit in
when
is an integer. The desired sum is
P.S. The 4 computational steps can be deduced through quick experimentation.
~ numerophile
Video Solutions
~ ThePuzzlr
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See Also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.