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 "2010 AIME II Problems/Problem 6"

(→‎Solution)
(→‎Solution)
Line 3: Line 3:
  
 
==Solution==
 
==Solution==
−
There are two ways for a monic fourth degree polynomial to be factored into two non-constant polynomials with real coefficients: into a cubic and a linear equation, or 2 [[quadratic]]s.
 
−
 
 
<br/>
 
<br/>
−
*'''Case 1''': The factors are cubic and linear.
+
You can always factor a polynomial into quadratic factors.
−
 
 
−
Let <math>x-r_1</math> be the linear root, where <math>r_1</math> is a root of the given quartic, and let <math>x^3+ax^2+bx+c</math> be the cubic.
 
−
 
 
−
By the [[Rational Root Theorem]], then <math>r_1=1,3,7, 9</math>, or <math>63</math>. Observe that
 
−
 
 
−
<cmath>(x^3+ax^2+bx+c)(x-r_1)=x^4+(a-r_1)x^3+(b-ar_1)x^2+(c-br_1)x-cr_1.</cmath>
 
−
 
 
−
Setting [[coefficient]]s equal, we have <math>a-r_1=0 \Longrightarrow a=r_1</math>, <math>b-ar_1=0\Longrightarrow b=a^2</math>, and <math>c-br_1=-n \Longrightarrow n=a^3-c</math>, and <math>-cr_1=63 \Longrightarrow c=\frac{-63}{a}</math>.
 
−
 
 
−
<br/>
 
−
 
 
−
It follows that <math>n=a^3+\frac{63}{a}</math>, <math>r_1=1,3,7, 9</math>, or <math>63</math>, which reach minimum when <math>r_1=3</math>, where <math>n=48</math>.
 
−
 
 
−
 
 
−
<br/>
 
−
*'''Case 2''': The factors are quadratics.
 
  
 
Let <math>x^2+ax+b</math> and <math>x^2+cx+d</math> be the two quadratics, so that
 
Let <math>x^2+ax+b</math> and <math>x^2+cx+d</math> be the two quadratics, so that

Revision as of 09:53, 21 November 2016

Problem

Find the smallest positive integer $n$ with the property that the polynomial $x^4 - nx + 63$ can be written as a product of two nonconstant polynomials with integer coefficients.

Solution


You can always factor a polynomial into quadratic factors.

Let $x^2+ax+b$ and $x^2+cx+d$ be the two quadratics, so that

\[(x^2 + ax + b )(x^2 + cx + d) = x^4 + (a + c)x^3 + (b + d + ac)x^2 + (ad + bc)x + bd.\]

Therefore, again setting coefficients equal, $a + c = 0\Longrightarrow a=-c$, $b + d + ac = 0\Longrightarrow b+d=a^2$ , $ad + bc = - n$, and so $bd = 63$.

Since $b+d=a^2$, the only possible values for $(b,d)$ are $(1,63)$ and $(7,9)$. From this we find that the possible values for $n$ are $\pm 8 \cdot 62$ and $\pm 4 \cdot 2$. Therefore, the answer is $\boxed{008}$.

See also

2010 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME 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=2010_AIME_II_Problems/Problem_6&oldid=81199"