2017 AMC 12A Problem 23

Question

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

\begin{aligned} &f(x)=x^4+x^3+bx^2 \\ &\quad {}+100x+c. \end{aligned}

What is

f(1)?

-9009

-8008

-7007

-6006

-5005

Since the three roots of

g

are roots of

f,

we have

f(x)=(x-q)\,g(x)

for one extra root

q

Match the

x^1

coefficients to find

q,

then

f(1)=(1-q)\,g(1)

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,

\begin{aligned} &f(x)=x^4+(a-q)x^3 \\ &\quad {}+(1-qa)x^2 \\ &\quad {}+(10-q)x-10q. \end{aligned}

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

\begin{aligned} &f(1)=(1-q)g(1) \\ &=91\cdot(-77)=-7007. \end{aligned}

Thus, the correct answer is C.

Problem 23 in other years