Difference between revisions of "2021 AMC 12B Problems/Problem 16"

Revision as of 18:42, 29 October 2022

Problem

Let $g(x)$ be a polynomial with leading coefficient $1,$ whose three roots are the reciprocals of the three roots of $f(x)=x^3+ax^2+bx+c,$ where $1<a<b<c.$ What is $g(1)$ in terms of $a,b,$ and $c?$

$\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}$

Solution 1

Note that $f(1/x)$ has the same roots as $g(x)$ , if it is multiplied by some monomial so that the leading term is $x^3$ they will be equal. We have $f(1/x) = \frac{1}{x^3} + \frac{a}{x^2}+\frac{b}{x} + c$ so we can see that $g(x) = \frac{x^3}{c}f(1/x)$ Therefore $g(1) = \frac{1}{c}f(1) = \boxed{\textbf{(A) }\frac{1+a+b+c}c}$

Solution 2 (Vieta's bash)

Let the three roots of $f(x)$ be $d$ , $e$ , and $f$ . (Here e does NOT mean 2.7182818...) We know that $a=-(d+e+f)$ , $b=de+ef+df$ , and $c=-def$ , and that $g(1)=1-\frac{1}{d}-\frac{1}{e}-\frac{1}{f}+\frac{1}{de}+\frac{1}{ef}+\frac{1}{df}-\frac{1}{def}$ (Vieta's). This is equal to $\frac{def-de-df-ef+d+e+f-1}{def}$ , which equals $\boxed{(\textbf{A}) \frac{1+a+b+c}{c}}$ . -dstanz5

Solution 3 (Fakesolve)

Because the problem doesn't specify what the coefficients of the polynomial actually are, we can just plug in any arbitrary polynomial that satisfies the constraints. Let's take $f(x) = (x+5)^3 = x^3+15x^2+75x+125$ . Then $f(x)$ has a triple root of $x = -5$ . Then $g(x)$ has a triple root of $-\frac{1}{5}$ , and it's monic, so $g(x) = \left(x+\frac{1}{5}\right)^3 = \frac{125x^3+75x^2+15x+1}{125}$ . We can see that this is $\frac{1+a+b+c}{c}$ , which is answer choice $\boxed{(A)}$ .

-Darren Yao

Solution 4

If we let $p, q,$ and $r$ be the roots of $f(x)$ , $f(x) = (x-p)(x-q)(x-r)$ and $g(x) = (x-\frac{1}{p})(x-\frac{1}{q})(x-\frac{1}{r})$ . The requested value, $g(1)$ , is then $(1-\frac{1}{p})(1-\frac{1}{q})(1-\frac{1}{r}) = \frac{(p-1)(q-1)(r-1)}{pqr}$ The numerator is $-f(1)$ (using the product form of $f(x)$ ) and the denominator is $-c$ , so the answer is $\frac{f(1)}{c} = \boxed{(\textbf{A}) \frac{1+a+b+c}{c}}$

- gting

Solution 5 (Good at Guessing)

The function $g(1) = \text{sum of coefficients}$ . If it's $(x-r)(x-s)(x-t)$ , then it becomes $(x-1/r)(x-1/s)(x-1/t).$ So, $-rst$ becomes $-1/rst$ , so $c$ becomes $1/c$ . Also, there is a $x^3$ so the answer must include $1$ . The only answer having both of these is $A$ .

~smellyman

-Extremelysupercooldude (minor latex edits and grammar)

Solution 6

It is well known that reversing the order of the coefficients of a polynomial turns each root into its corresponding reciprocal. Thus, a polynomial with the desired roots may be written as $cx^3+bx^2+a+1$ . As the problem statement asks for a monic polynomial, our answer is $\boxed{(\textbf{A}) \frac{1+a+b+c}{c}}$

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=vCEJzhDRUoU

Video Solution by OmegaLearn (Vieta's Formula)

https://youtu.be/afrGHNo_JcY

Video Solution by Hawk Math

https://www.youtube.com/watch?v=p4iCAZRUESs

@@ Line 31: / Line 31: @@
 - gting
-==Solution 5 (good at guessing)==
+==Solution 5 (Good at Guessing)==
-g(1) = sum of coefficients. if its (x-r)(x-s)(x-t), then it becomes (x-1/r)(x-1/s)(x-1/t)
+The function <math>g(1) = \text{sum of coefficients}</math>. If it's <math>(x-r)(x-s)(x-t)</math>, then it becomes <math>(x-1/r)(x-1/s)(x-1/t).</math>
-so -rst becomes -1/rst so c becomes 1/c. Also, there is x^3 so the answer must include 1. the only answer having both of these is A.
+So, <math>-rst</math> becomes <math>-1/rst</math>, so <math>c</math> becomes <math>1/c</math>. Also, there is a <math>x^3</math> so the answer must include <math>1</math>. The only answer having both of these is <math>A</math>.
 ~smellyman
+-Extremelysupercooldude (minor latex edits and grammar)
 ==Solution 6==

2021 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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