2021 AMC 12B Problems/Problem 16: Difference between revisions

Latest revision as of 09:02, 1 October 2023

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 $g(x) = \left(x-\frac{1}{p}\right)\left(x-\frac{1}{q}\right)\left(x-\frac{1}{r}\right)$ . The requested value, $g(1)$ , is then $\left(1-\frac{1}{p}\right)\left(1-\frac{1}{q}\right)\left(1-\frac{1}{r}\right) = \frac{(p-1)(q-1)(r-1)}{pqr}$ 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 $(x-\dfrac{1}{r})(x-\dfrac{1}{s})(x-\dfrac{1}{t}).$ So, $-rst$ becomes $-\dfrac{1}{rst}$ , so $c$ becomes $\dfrac{1}{c}$ . 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 (🚀Under 2 min 🚀)

https://youtu.be/vPw6VxuZvQU

~Education, the Study of Everything

Video Solution by OmegaLearn

https://youtu.be/M4Ffhp9NLKY?t=923

~ pi_is_3.14

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 24: / Line 24: @@
 ==Solution 4==
-If we let <math>p, q, </math> and <math>r</math> be the roots of <math>f(x)</math>, <math>f(x) = (x-p)(x-q)(x-r)</math> and <math>g(x) = (x-\frac{1}{p})(x-\frac{1}{q})(x-\frac{1}{r})</math>. The requested value, <math>g(1)</math>, is then
+If we let <math>p, q, </math> and <math>r</math> be the roots of <math>f(x)</math>, <math>f(x) = (x-p)(x-q)(x-r)</math> and <math>g(x) = \left(x-\frac{1}{p}\right)\left(x-\frac{1}{q}\right)\left(x-\frac{1}{r}\right)</math>. The requested value, <math>g(1)</math>, is then
-<cmath>(1-\frac{1}{p})(1-\frac{1}{q})(1-\frac{1}{r}) = \frac{(p-1)(q-1)(r-1)}{pqr}</cmath>
+<cmath>\left(1-\frac{1}{p}\right)\left(1-\frac{1}{q}\right)\left(1-\frac{1}{r}\right) = \frac{(p-1)(q-1)(r-1)}{pqr}</cmath>
 The numerator is <math>-f(1)</math> (using the product form of <math>f(x)</math> ) and the denominator is <math>-c</math>, so the answer is
 <cmath>\frac{f(1)}{c} = \boxed{(\textbf{A}) \frac{1+a+b+c}{c}}</cmath>
 - gting
+==Solution 5 (Good at Guessing)==
+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-\dfrac{1}{r})(x-\dfrac{1}{s})(x-\dfrac{1}{t}).</math>
+So, <math>-rst</math> becomes <math>-\dfrac{1}{rst}</math>, so <math>c</math> becomes <math>\dfrac{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==
+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 <math>cx^3+bx^2+a+1</math>. As the problem statement asks for a monic polynomial, our answer is <cmath>\boxed{(\textbf{A}) \frac{1+a+b+c}{c}}</cmath>
+==Video Solution (🚀Under 2 min 🚀)==
+https://youtu.be/vPw6VxuZvQU
+<i>~Education, the Study of Everything</i>
+== Video Solution by OmegaLearn ==
+https://youtu.be/M4Ffhp9NLKY?t=923
+~ pi_is_3.14
 ==Video Solution by Punxsutawney Phil==
@@ Line 36: / Line 58: @@
 == Video Solution by OmegaLearn (Vieta's Formula) ==
 https://youtu.be/afrGHNo_JcY
-~ pi_is_3.14
 ==Video Solution by Hawk Math==
@@ Line 44: / Line 64: @@
 ==See Also==
 {{AMC12 box|year=2021|ab=B|num-b=15|num-a=17}}
-{{AMC10 box|year=2021|ab=B|num-b=17|num-a=19}}
+[[Category:Intermediate Algebra Problems]]
 {{MAA Notice}}

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