Difference between revisions of "2018 UNCO Math Contest II Problems/Problem 8"
(→Solution) |
(→Solution) |
||
Line 5: | Line 5: | ||
Let <math>p(x) = x^{2018} + x^{1776}-3x^4-3</math>. Find the remainder when you divide <math>p(x)</math> by <math>x^3-x</math> | Let <math>p(x) = x^{2018} + x^{1776}-3x^4-3</math>. Find the remainder when you divide <math>p(x)</math> by <math>x^3-x</math> | ||
− | By the division algorithm, let <math>p(x)\: | + | By the division algorithm, let <math>p(x)\: = x^{2018} + x^{1776}-3x^4-3 {\equiv}\: (x^3 - x)Q(x) + (Ax^2 + Bx + C)</math> for some quotient <math>Q(x)</math> |
Consider the roots of <math>x^3-x=0 \implies x=0\: or\: x=1 \:or\: x=-1</math> | Consider the roots of <math>x^3-x=0 \implies x=0\: or\: x=1 \:or\: x=-1</math> |
Revision as of 20:17, 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 |