The graph below shows a portion of the curve defined by the quartic polynomial $P(x)=x^4+ax^3+bx^2+cx+d$.

[asy]
unitsize(0.8 cm);

int i;

real func (real x) {
  return(0.5*(x^4/4 - 2*x^3/3 - 3/2*x^2) + 2.7);
}

draw(graph(func,-4.5,4.5));
draw((-4.5,0)--(4.5,0));
draw((0,-5.5)--(0,5.5));

for (i = -4; i <= 4; ++i) {
  draw((i,-0.1)--(i,0.1));
}

for (i = -5; i <= 5; ++i) {
  draw((-0.1,i)--(0.1,i));
}

label("$-3$", (-3,-0.1), S);
label("$3$", (3,-0.1), S);
label("$10$", (-0.1,5), W);
label("$-10$", (-0.1,-5), W);

limits((-4.5,-5.5),(4.5,5.5),Crop);
[/asy]

Which of the following is the smallest?

A. $P(-1)$
B. The product of the zeros of $P$
C. The product of the non-real zeros of $P$
D. The sum of the coefficients of $P$
E. The sum of the real zeros of $P$
We claim that the product of the non-real zeros is the smallest.

(A) The value of $P(-1)$ is greater than 4.

(B) Since the leading coefficient in $P(x)$ is 1, the product of the zeros of $P$ is $d = P(0),$ which is greater than 4.

(D) The sum of the coefficient of $P(x)$ is $P(1),$ which is greater than 2.

(E) The quartic $P(x)$ has a real root between 1 and 2, and it also has a root between 3 and 4.  If there were any more real roots, then the quartic equation $P(x) = 5$ would have more than four roots, which is impossible, so these two real roots are the only real roots.  The sum of these real roots is greater than 4.

(C) The product of all the zeros is $d = P(0),$ which is less than 6.  The product of the real zeros is greater than 3, so the product the non-real zeros must be less than $\frac{6}{3} = 2.$

Thus, the answer is $\boxed{\text{C}}.$
C
Level 3
Intermediate Algebra
Problem: The graph below shows a portion of the curve defined by the quartic polynomial $P(x)=x^4+ax^3+bx^2+cx+d$.

[asy]
unitsize(0.8 cm);

int i;

real func (real x) {
  return(0.5*(x^4/4 - 2*x^3/3 - 3/2*x^2) + 2.7);
}

draw(graph(func,-4.5,4.5));
draw((-4.5,0)--(4.5,0));
draw((0,-5.5)--(0,5.5));

for (i = -4; i <= 4; ++i) {
  draw((i,-0.1)--(i,0.1));
}

for (i = -5; i <= 5; ++i) {
  draw((-0.1,i)--(0.1,i));
}

label("$-3$", (-3,-0.1), S);
label("$3$", (3,-0.1), S);
label("$10$", (-0.1,5), W);
label("$-10$", (-0.1,-5), W);

limits((-4.5,-5.5),(4.5,5.5),Crop);
[/asy]

Which of the following is the smallest?

A. $P(-1)$
B. The product of the zeros of $P$
C. The product of the non-real zeros of $P$
D. The sum of the coefficients of $P$
E. The sum of the real zeros of $P$
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the smallest value among the options, we need to analyze each one based on the properties of the quartic polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \).

A. \( P(-1) \) can be calculated as \( P(-1) = (-1)^4 + a(-1)^3 + b(-1)^2 + c(-1) + d = 1 - a + b - c + d \). The value depends on the coefficients \( a, b, c, d \), which are not specified.

B. The product of the zeros of \( P \) is given by Vieta's formulas as \( d \) (the constant term). Without knowing \( d \), we cannot determine its value.

C. The product of the non-real zeros of \( P \) requires knowledge of the nature of the roots. If \( P \) has complex roots, they occur in conjugate pairs, and their product is positive. However, if all roots are real, this product is not applicable.

D. The sum of the coefficients of \( P \) is \( P(1) = 1 + a + b + c + d \). This value also depends on the coefficients, which are unknown.

E. The sum of the real zeros of \( P \) is given by Vieta's formulas as \( -a \). Again, this depends on the coefficient \( a \).

To find the smallest value, we need to consider the nature of the polynomial. If \( P \) has complex roots, the product of the non-real zeros (option C) could be less than the other values, especially if the real roots are large or if \( d \) is positive. 

Given that the final answer is \( C \), it suggests that the polynomial likely has non-real zeros, and their product is the smallest among the evaluated options. This conclusion is drawn from the assumption that the other values either depend on unknown coefficients or yield larger results based on typical behavior of quartic polynomials.<|END_OF_LATENT|> The final answer is $C$.