2017 AMC 12A Problem 23

Below is the professionally curated solution for Problem 23 of the 2017 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12A solutions, or check the answer key.

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Concepts:polynomial factoring

Difficulty rating: 2380

23.

For certain real numbers $a,$ $b,$ and $c,$ the polynomial $g(x)=x^3+ax^2+x+10$ has three distinct roots, and each root of $g(x)$ is also a root of the polynomial $f(x)=x^4+x^3+bx^2+100x+c.$ What is $f(1)?$

$-9009$

$-8008$

$-7007$

$-6006$

$-5005$

Solution:

Since $g$ has three distinct roots all shared by the quartic $f,$ we can write $f(x)=(x-q)g(x)$ for some remaining root $q.$ Expanding, $f(x)=x^4+(a-q)x^3+(1-qa)x^2+(10-q)x-10q.$

Matching the $x$ coefficient, $10-q=100,$ so $q=-90.$ Matching the $x^3$ coefficient, $a-q=1,$ so $a=-89.$

Then $g(1)=1+a+1+10=12-89=-77$ and $1-q=91,$ so $f(1)=(1-q)g(1)=91\cdot(-77)=-7007.$

Thus, the correct answer is C.

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