The graph below shows a portion of the curve defined by the quartic polynomial P(x)=x⁴ +ax³ +bx² +cx+d. Which of the following is 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

(I'm not really even sure what B through E mean)

[x=0 at around 5]


Any hints or answers are greatly appreciated.

Do you know about complex numbers? If you do, this can be shorter so I'll leave it for now.

Tom.

It helps to know that the "zeros" of a function are the roots. That is, the values x for which f(x) = 0.

I'll give you a few more hints.

Michael's already pointed out what "zeros" means. You indicated that you don't know what D means - the coefficients of a polynomial are the numbers which multiply each power of X (or whatever the variable is), so in this case the coefficients of P are 1, a, b, c, and d.

So now you can work out A and D. For the other three, here's a bit of useful theory:

I'll assume that you're at least aware of what complex numbers are - if not please correct me! There is a theorem which states that any polynomial f of degree n (that is, the highest power of X is Xⁿ ) can be factorised completely as a product of linear factors, that means

f(X) = Xⁿ + b_n-1 X^n-1 + ... + b₁ X + b₀ = (X - a₁ )(X - a₂ )...(x - a_n )

for some complex numbers a_i , which are the zeros of f. This is called the Fundamental Theorem of Algebra, by the way.

Anyway, what's useful about this for our purposes is that you can then multiply out the brackets, and you find that

b₀ = (-1)ⁿ a₁ a₂ ...a_n

so the product of the roots is (-1)ⁿ b₀ , and also

b_n-1 = -(a₁ + a₂ + ... + a_n )

so the sum of the roots is -b_n-1 , which might help you a bit.

Another (possibly) useful thing to notice is that from the graph, and with a bit of differentiation b < 0, c < 0 and d > 0.

I'm stuck now I'm afraid, and it's too late for me to be making any sense, so I'll go now.

James.

Try considering that if the roots are a, b, g, d then the quartic can be written as (x-a)(x-b)(x-g)(x-d) = x⁴+a x³+b x²+c x+d=0. Can you then find expressions for the product of the roots and the sum of the roots in terms of a, b, c and d?
Tom.