2017 AMC 10A Problem 24

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 \\ &\quad {}+ bx^2 + 100x + c. \end{aligned}

What is

f(1)?

-9009

-8008

-7007

-6006

-5005

Because the roots of

g

are roots of

f

, factor

f

by

g

Compare the

x^3

and

x

coefficients to find the extra root and

a

Because the three distinct roots of the cubic

g

are also roots of the monic quartic

f,

we can write

f(x)=g(x)(x-r)

for some real number

r.

The coefficient of

x

in this product is

10-r,

so

10-r=100

and

r=-90.

The coefficient of

x^3

is

a-r,

so

a-r=1

and

a=-89.

Therefore,

\begin{aligned} f(1)&=g(1)(1-r)\\ &=(1-89+1+10)(91)\\ &=(-77)(91)\\ &=-7007. \end{aligned}

Thus, C is the correct answer.

Problem 24 in other years