jmc

Question

The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function

f (x) = a x^{2} + b x + c

are equal. Their common value must also be which of the following?

(A) The coefficient of

x^{2} .

(B) The coefficient of

x .

(C) The

y

-intercept of the graph of

y = f (x) .

(D) One of the

x

-intercepts of the graph of

y = f (x) .

(E) The mean of the

x

-intercepts of the graph of

y = f (x) .

Accepted Answer

By Vieta's formulas, the sum of the zeros is

- \frac{b}{a},

and the sum of the coefficients is

\frac{c}{a},

so

b = - c .

Then the sum of the coefficients is

a + b + c = a,

which is the coefficient of

x^{2} .

Thus, the answer is

(A) .

To see that none of the other choices can work, consider

f (x) = - 2 x^{2} - 4 x + 4.

The sum of the zeros, the product of the zero, and the sum of the coefficients are all

- 2.

The coefficient of

x

is

- 4,

the

y

-intercept of the graph of

y = f (x)

is 4, the

x

-intercepts are

- 1 \pm 3,

and the mean of the

x

-intercepts is

- 1,

so none of the other choices work.