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 "1969 AHSME Problems/Problem 34"

(→‎Solution)
(→‎Solution)
Line 13: Line 13:
 
Let the polynomial <math>Q(x)</math> be the quotient when <math>x^{100}</math> is divided by <math>x^2-3x+2</math>, and let <math>R</math> the remainder be <math>ax+b</math>, for some real <math>a</math> and <math>b</math>. Then we can write: <math>x^100=(x^2-3x+2)Q(x)+ax+b</math>. Since it is hard to deal with <math>Q(x)</math> (it is of degree 98!), we factor <math>x^2-3x+2</math> as <math>(x-2)(x-1)</math> so we can eliminate <math>Q(x)</math> by plugging in <math>x</math> values of <math>2</math> and <math>1</math>.
 
Let the polynomial <math>Q(x)</math> be the quotient when <math>x^{100}</math> is divided by <math>x^2-3x+2</math>, and let <math>R</math> the remainder be <math>ax+b</math>, for some real <math>a</math> and <math>b</math>. Then we can write: <math>x^100=(x^2-3x+2)Q(x)+ax+b</math>. Since it is hard to deal with <math>Q(x)</math> (it is of degree 98!), we factor <math>x^2-3x+2</math> as <math>(x-2)(x-1)</math> so we can eliminate <math>Q(x)</math> by plugging in <math>x</math> values of <math>2</math> and <math>1</math>.
  
−
<math>x^100=(x-2)(x-1)Q(x)+ax+b</math>
+
<math>x^{100}=(x-2)(x-1)Q(x)+ax+b</math>
−
<math>2^100=(2-2)(x-1)Q(x)+ax+b</math>
+
<math>2^{100}=(2-2)(x-1)Q(x)+ax+b</math>
−
<math>2^100=2a+b</math>.
+
<math>2^{100}=2a+b</math>.
  
−
Similarly, <math>1^100=a+b</math>.
+
Similarly, <math>1^{100}=a+b</math>.
 
Solving this system of equations gives <math>a=2^{100}-1</math> and <math>b=2-2^{100}</math>. Thus, <math>R=ax+b=(2^{100}-1)x+(2-2^{100})</math>. Expanding and combining <math>x</math> terms, we see that the answer is <math>\fbox{B}</math>.
 
Solving this system of equations gives <math>a=2^{100}-1</math> and <math>b=2-2^{100}</math>. Thus, <math>R=ax+b=(2^{100}-1)x+(2-2^{100})</math>. Expanding and combining <math>x</math> terms, we see that the answer is <math>\fbox{B}</math>.
  

Revision as of 15:54, 10 July 2015

Problem

The remainder $R$ obtained by dividing $x^{100}$ by $x^2-3x+2$ is a polynomial of degree less than $2$. Then $R$ may be written as:


$\text{(A) }2^{100}-1 \quad \text{(B) } 2^{100}(x-1)-(x-2)\quad \text{(C) } 2^{200}(x-3)\quad\\ \text{(D) } x(2^{100}-1)+2(2^{99}-1)\quad \text{(E) } 2^{100}(x+1)-(x+2)$

Solution

Let the polynomial $Q(x)$ be the quotient when $x^{100}$ is divided by $x^2-3x+2$, and let $R$ the remainder be $ax+b$, for some real $a$ and $b$. Then we can write: $x^100=(x^2-3x+2)Q(x)+ax+b$. Since it is hard to deal with $Q(x)$ (it is of degree 98!), we factor $x^2-3x+2$ as $(x-2)(x-1)$ so we can eliminate $Q(x)$ by plugging in $x$ values of $2$ and $1$.

$x^{100}=(x-2)(x-1)Q(x)+ax+b$ $2^{100}=(2-2)(x-1)Q(x)+ax+b$ $2^{100}=2a+b$.

Similarly, $1^{100}=a+b$. Solving this system of equations gives $a=2^{100}-1$ and $b=2-2^{100}$. Thus, $R=ax+b=(2^{100}-1)x+(2-2^{100})$. Expanding and combining $x$ terms, we see that the answer is $\fbox{B}$.

See also

1969 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 33 		Followed by
Problem 35
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
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=1969_AHSME_Problems/Problem_34&oldid=71086"