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 $f(1)?$

$-9009$

$-8008$

$-7007$

$-6006$

$-5005$

Answer: C

Concepts:polynomial factoring

Difficulty rating: 2110

Solution:

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.

2017 AMC 10A Problem 24

Answer: C

Difficulty rating: 2110

Solution:

Problem 24 in Other Years