Difference between revisions of "2002 AMC 10P Problems/Problem 11"

Revision as of 17:46, 14 July 2024

Let $P(x)=kx^3 + 2k^2x^2+k^3.$ Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x).$

$\text{(A) }-8 \qquad \text{(B) }-4 \qquad \text{(C) }0 \qquad \text{(D) }5 \qquad \text{(E) }8$

2002 AMC 10P (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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
All AMC 10 Problems and Solutions

@@ Line 1: / Line 1: @@
-== Problem 12 ==
+== Problem ==
-For <math>f_n(x)=x^n</math> and <math>a \neq 1</math> consider
+Let <math>P(x)=kx^3 + 2k^2x^2+k^3.</math> Find the sum of all real numbers <math>k</math> for which <math>x-2</math> is a factor of <math>P(x).</math>
-<math>\text{I. } (f_{11}(a)f_{13}(a))^{14}</math>
-<math>\text{II. } f_{11}(a)f_{13}(a)f_{14}(a)</math>
-<math>\text{III. } (f_{11}(f_{13}(a)))^{14}</math>
-<math>\text{IV. } f_{11}(f_{13}(f_{14}(a)))</math>
-Which of these equal <math>f_{2002}(a)?</math>
 <math>
-\text{(A) I and II only}
+\text{(A) }-8
 \qquad
-\text{(B) II and III only}
+\text{(B) }-4
 \qquad
-\text{(C) III and IV only}
+\text{(C) }0
 \qquad
-\text{(D) II, III, and IV only}
+\text{(D) }5
 \qquad
-\text{(E) all of them}
+\text{(E) }8
 </math>