Difference between revisions of "2018 UNCO Math Contest II Problems/Problem 8"
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For <math>p(0)</math>, | For <math>p(0)</math>, | ||
− | <math>(0)^{2018} + (0)^{1776}-3(0)^4-3 = ((0)^3 - (0))Q( | + | <math>(0)^{2018} + (0)^{1776}-3(0)^4-3 = ((0)^3 - (0))Q(0) + (A(0)^2 + B(0) + C)</math> |
<math>C=-3</math> | <math>C=-3</math> | ||
Line 17: | Line 17: | ||
For <math>p(1)</math>, | For <math>p(1)</math>, | ||
− | <math>(1)^{2018} + (1)^{1776}-3(1)^4-3 = ((1)^3 - (1))Q( | + | <math>(1)^{2018} + (1)^{1776}-3(1)^4-3 = ((1)^3 - (1))Q(1) + (A(1)^2 + B(1) + C)</math> |
<math>1 + 1 - 3 -3 = A + B - 3</math> | <math>1 + 1 - 3 -3 = A + B - 3</math> | ||
Line 23: | Line 23: | ||
<math>A+B=-1</math> | <math>A+B=-1</math> | ||
− | For <math>p(-1)</math> | + | For <math>p(-1)</math>, |
− | <math>(-1)^{2018} + (-1)^{1776}-3(-1)^4-3 = ((-1)^3 - (-1))Q( | + | <math>(-1)^{2018} + (-1)^{1776}-3(-1)^4-3 = ((-1)^3 - (-1))Q(-1) + (A(-1)^2 + B(-1) + C)</math> |
<math>1 + 1 -3 -3 = A - B + -3</math> | <math>1 + 1 -3 -3 = A - B + -3</math> |
Latest revision as of 20:19, 26 September 2022
Problem
Let . Find the remainder when you divide
by
Solution
Let . Find the remainder when you divide
by
By the division algorithm, let for some quotient
Consider the roots of
For ,
For ,
For ,
Solving, we get and
the remainder is
See also
2018 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |