2001 AMC 12 Problem 19

Question

The polynomial

P(x) = x^3 + ax^2 + bx + c

has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the

y

-intercept of the graph of

y = P(x)

is

2,

what is

b?

-11

-10

-9

1

5

By Vieta, the sum of the zeros is

-a

and the product is

-c;

the sum of coefficients is

P(1)

The

y

-intercept gives

c = 2,

so the common value is the product of the zeros

The

y

-intercept is

P(0) = c = 2.

By Vieta’s formulas the product of the zeros is

-c = -2,

the mean of the zeros is

-\dfrac{a}{3},

and the sum of the coefficients is

P(1) = 1 + a + b + c.

All three are equal to

-2.

From

-\dfrac{a}{3} = -2

we get

a = 6.

Then

1 + a + b + c = -2

becomes

1 + 6 + b + 2 = -2,

so

b = -11.

Thus, the correct answer is A.

Problem 19 in other years