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 "2007 iTest Problems/Problem 18"

(Created page with "== Problem == Suppose that <math>x^3+px^2+qx+r</math> is a cubic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real num...")
 
m (→‎Problem)
 
(3 intermediate revisions by 2 users not shown)
Line 8: Line 8:
 
\text{(C) }108\qquad
 
\text{(C) }108\qquad
 
\text{(D) It could be }0 \text{ or } 4\qquad
 
\text{(D) It could be }0 \text{ or } 4\qquad
−
\text{(E) It could be }0 \text{ or } 108 \\ </math>
+
\text{(E) It could be }0 \text{ or } 108</math>
  
 
<math>\text{(F) }18\qquad
 
<math>\text{(F) }18\qquad
 
\text{(G) }-4\qquad
 
\text{(G) }-4\qquad
 
\text{(H) }-108\qquad
 
\text{(H) }-108\qquad
−
\text{(I) It could be } 0 \text{ or } -4 \\ </math>
+
\text{(I) It could be } 0 \text{ or } -4</math>
  
 
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad
 
<math>\text{(J) It could be } 0 \text{ or } {-108} \qquad
 
\text{(K) It could be } 4 \text{ or } {-4}\qquad
 
\text{(K) It could be } 4 \text{ or } {-4}\qquad
−
\text{(L) There is no such value of } r\qquad \\ </math>
+
\text{(L) There is no such value of } r\qquad</math>
  
 
<math>\text{(M) } 1 \qquad
 
<math>\text{(M) } 1 \qquad
 
\text{(N) } {-2} \qquad  
 
\text{(N) } {-2} \qquad  
 
\text{(O)  It could be } 4 \text{ or } -4 \qquad
 
\text{(O)  It could be } 4 \text{ or } -4 \qquad
−
\text{(P)  It could be } 0 \text{ or } -2 \qquad \\ </math>
+
\text{(P)  It could be } 0 \text{ or } -2 \qquad</math>
  
 
<math>\text{(Q)  It could be } 2007 \text{ or a yippy dog} \qquad
 
<math>\text{(Q)  It could be } 2007 \text{ or a yippy dog} \qquad
−
\text{(R)  } 2007 \\ </math>
+
\text{(R)  } 2007</math>
  
−
== Solution  ==
+
==Solutions==
 +
 
 +
===Solution 1===
 +
 
 +
By [[Vieta's Formulas]],
 +
<cmath>2a+b = 6</cmath>
 +
<cmath>2ab + a^2 = 9</cmath>
 +
From the first equation, <math>b = -2a+6</math>.  Substitute into the second equation to get
 +
<cmath>-4a^2 + 12a + a^2 = 9</cmath>
 +
<cmath>-3a^2 + 12a - 9 = 0</cmath>
 +
<cmath>a^2 - 4a + 3 = 0</cmath>
 +
Thus, <math>a = 3</math> or <math>a = 1</math>.  If <math>a = 3</math>, then <math>b = 0</math>, so by Vieta's Formulas, <math>r = 0</math>.  If <math>a = 1</math>, then <math>b = 4</math>, so by Vieta's Formulas, <math>r = -4</math>.  The answer is <math>\boxed{\textbf{(I)}}</math>.
 +
 
 +
===Solution 2===
 +
 
 +
The equation of the [[polynomial]] with double root <math>a</math> and single root <math>b</math> is
 +
<cmath>(x-a)^2 (x-b)</cmath>
 +
Expanding the polynomial results in
 +
<cmath>(x^2 - 2ax + a^2)(x-b)</cmath>
 +
<cmath>x^3 - (2a+b)x^2 + (2ab+a^2)x - a^2b</cmath>
 +
Since <math>p = -6</math> and <math>q = 9</math>, we can write a [[system of equations]].
 +
<cmath>2a+b = 6</cmath>
 +
<cmath>2ab + a^2 = 9</cmath>
 +
Solving for <math>a</math> results in <math>a = 3</math> or <math>a = 1</math>.  If <math>a = 3</math>, then <math>b = 0</math>, so <math>r = 0</math>. If <math>a = 1</math>, then <math>b = 4</math>, so <math>r = -4</math>.  Thus, the answer is <math>\boxed{\textbf{(I)}}</math>.
 +
 
 +
==See Also==
 +
{{iTest box|year=2007|num-b=17|num-a=19}}
 +
 
 +
[[Category:Introductory Algebra Problems]]

Latest revision as of 17:01, 10 June 2018

Problem

Suppose that $x^3+px^2+qx+r$ is a cubic with a double root at $a$ and another root at b, where $a$ and $b$ are real numbers. If $p=-6$ and $q=9$, what is $r$?

$\text{(A) }0\qquad \text{(B) }4\qquad \text{(C) }108\qquad \text{(D) It could be }0 \text{ or } 4\qquad \text{(E) It could be }0 \text{ or } 108$

$\text{(F) }18\qquad \text{(G) }-4\qquad \text{(H) }-108\qquad \text{(I) It could be } 0 \text{ or } -4$

$\text{(J) It could be } 0 \text{ or } {-108} \qquad \text{(K) It could be } 4 \text{ or } {-4}\qquad \text{(L) There is no such value of } r\qquad$

$\text{(M) } 1 \qquad \text{(N) } {-2} \qquad \text{(O) It could be } 4 \text{ or } -4 \qquad \text{(P) It could be } 0 \text{ or } -2 \qquad$

$\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad \text{(R) } 2007$

Solutions

Solution 1

By Vieta's Formulas, \[2a+b = 6\] \[2ab + a^2 = 9\] From the first equation, $b = -2a+6$. Substitute into the second equation to get \[-4a^2 + 12a + a^2 = 9\] \[-3a^2 + 12a - 9 = 0\] \[a^2 - 4a + 3 = 0\] Thus, $a = 3$ or $a = 1$. If $a = 3$, then $b = 0$, so by Vieta's Formulas, $r = 0$. If $a = 1$, then $b = 4$, so by Vieta's Formulas, $r = -4$. The answer is $\boxed{\textbf{(I)}}$.

Solution 2

The equation of the polynomial with double root $a$ and single root $b$ is \[(x-a)^2 (x-b)\] Expanding the polynomial results in \[(x^2 - 2ax + a^2)(x-b)\] \[x^3 - (2a+b)x^2 + (2ab+a^2)x - a^2b\] Since $p = -6$ and $q = 9$, we can write a system of equations. \[2a+b = 6\] \[2ab + a^2 = 9\] Solving for $a$ results in $a = 3$ or $a = 1$. If $a = 3$, then $b = 0$, so $r = 0$. If $a = 1$, then $b = 4$, so $r = -4$. Thus, the answer is $\boxed{\textbf{(I)}}$.

See Also

2007 iTest (Problems, Answer Key)
Preceded by:
Problem 17 		Followed by:
Problem 19
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 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems/Problem_18&oldid=95045"