G.MATH
POLYNOMIALS AND THEIR ZEROS AMC
Polynomials 2001 AMC12 #19

The polynomial  has the property that the average of its zeros, the product of its zeros, and the sum of its coefficients are all equal. The y-intercept of the graph of  is 2. What is b?
VIDEO LECTURE


MAIN CONCEPTS
Filter information from the polynomial and use the zero-coefficient relationship to solve.
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COMMENTS
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SOLUTION
In order to find b, it seems obvious that we may need to find the other unknowns a and c. The strategy here is to use the information we have to find these unknowns in a systematic manner. It is also advise that the student be familiar with the zero-coefficient relationships which are so important in all types of polynomials questions.

Basically, our information of this polynomials is

1. The average of its zeros, the product of its zeros, and the sum of its coefficients are all equal.
2. The y-intercept is 2.

After some thinking we can easily get c from information number 2. By putting x = 0, which is where the y-intercept occurs, we have c = P (0) = 2. Hence c = 2.

Our polynomials is now . Mathematical, point number 1 is

Here, we have to employ the zero-coefficient relationships to advance further. It allows us to express the terms  in a, b, and c, which leads to the solution. Recall that for a polynomial of the 3rd degree

where  are the first, second and third roots and taking note of the negative sign first. Substituting this into the previous equation,

which makes it a breeze to solve.

We just use the other equation and solve for b.

The answer is -11. I advise that the student be familiar with the zero-coefficient relationship for the general polynomial.

For a polynomial of the form

The coefficients and related to the zeros by

or in general

All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
Video courtesy of YouTube.com service.
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