Difference between revisions of "1977 AHSME Problems/Problem 28"
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==Problem== | ==Problem== | ||
+ | Let <math>g(x)=x^5+x^4+x^3+x^2+x+1</math>. What is the remainder when the polynomial <math>g(x^{12})</math> is divided by the polynomial <math>g(x)</math>? | ||
+ | <math>\textbf{(A) }6\qquad | ||
+ | \textbf{(B) }5-x\qquad | ||
+ | \textbf{(C) }4-x+x^2\qquad | ||
+ | \textbf{(D) }3-x+x^2-x^3\qquad \ | ||
+ | \textbf{(E) }2-x+x^2-x^3+x^4</math> | ||
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<cmath>g(x) = 0</cmath> | <cmath>g(x) = 0</cmath> | ||
We have <math>5</math> values of <math>x</math> that return <math>R(x) = 6</math>. However, <math>g(x)</math> is quintic, implying the remainder is of degree <math>4</math> — contradicted by the <math>5</math> solutions. Thus, the only remaining possibility is that the remainder is a constant <math>\boxed{6}</math>. | We have <math>5</math> values of <math>x</math> that return <math>R(x) = 6</math>. However, <math>g(x)</math> is quintic, implying the remainder is of degree <math>4</math> — contradicted by the <math>5</math> solutions. Thus, the only remaining possibility is that the remainder is a constant <math>\boxed{6}</math>. | ||
+ | |||
+ | == See also == | ||
+ | {{AHSME box | year = 1977 | num-b = 27 | after = 29}} |
Revision as of 19:21, 31 October 2021
Contents
[hide]Problem
Let . What is the remainder when the polynomial is divided by the polynomial ?
Solution 1
Let be the remainder when is divided by . Then is the unique polynomial such that is divisible by , and .
Note that is a multiple of . Also, Each term is a multiple of . For example, Hence, is a multiple of , which means that is a multiple of . Therefore, the remainder is . The answer is (A).
Solution 2
We express the quotient and remainder as follows. Note that the solutions to correspond to the 6th roots of unity, excluding . Hence, we have , allowing us to set: We have values of that return . However, is quintic, implying the remainder is of degree — contradicted by the solutions. Thus, the only remaining possibility is that the remainder is a constant .
See also
1977 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by 29 | |
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 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |