Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1965 AHSME Problems/Problem 27"

m (formatting changes, typo fixes)
(Added solution to the problem)
 
Line 25: Line 25:
  
 
Thus, the answer is <math>\boxed{\textbf{(A) }0}</math>.
 
Thus, the answer is <math>\boxed{\textbf{(A) }0}</math>.
 +
 +
 +
== Solution 2 ==
 +
 +
As in the other solution, let <math>h(y)=y^2+my+2</math>. By dividing <math>h(y)</math> by <math>(y-1)</math> (through either long or [[synthetic division|synthetic]] division), we get a remainder <math>R_1=m+3</math>. Similarly, dividing <math>h(y)</math> by <math>(y+1)</math> yields the remainder <math>R_2=3-m</math>. Setting <math>R_1=R_2</math>, we see that <math>m+3=3-m</math>, which only holds when <math>m=0</math>, which corresponds to answer choice <math>\fbox{\textbf{(A)}}</math>.
  
 
== See Also ==
 
== See Also ==

Latest revision as of 19:21, 18 July 2024

Problem

When $y^2 + my + 2$ is divided by $y - 1$ the quotient is $f(y)$ and the remainder is $R_1$. When $y^2 + my + 2$ is divided by $y + 1$ the quotient is $g(y)$ and the remainder is $R_2$. If $R_1 = R_2$ then $m$ is:

$\textbf{(A)}\ 0 \qquad \textbf{(B) }\ 1 \qquad \textbf{(C) }\ 2 \qquad \textbf{(D) }\ - 1 \qquad \textbf{(E) }\ \text{an undetermined constant}$

Solution

Let $h(y)=y^2+my+2$

$h(y)=y^2+my+2=f(y)(y-1)+R_1$

$h(y)=y^2+my+2=g(y)(y+1)+R_2$

$h(1)=3+m=R_1$

$h(-1)=3-m=R_2$

$m=0$

Thus, the answer is $\boxed{\textbf{(A) }0}$.


Solution 2

As in the other solution, let $h(y)=y^2+my+2$. By dividing $h(y)$ by $(y-1)$ (through either long or synthetic division), we get a remainder $R_1=m+3$. Similarly, dividing $h(y)$ by $(y+1)$ yields the remainder $R_2=3-m$. Setting $R_1=R_2$, we see that $m+3=3-m$, which only holds when $m=0$, which corresponds to answer choice $\fbox{\textbf{(A)}}$.

See Also

1965 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 26 		Followed by
Problem 28
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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1965_AHSME_Problems/Problem_27&oldid=223064"