The Art and Craft of Problem Solving

168 CHAPTER 5 ALGEBRA

Relationship Between Zeros and Coefficients

If we multiply out the right-hand side of equation (2) on page 166, we can get a series

of expressions for the coefficients of the polynomial in terms of its zeros. This seems

like a pretty complicated and tedious job, so let us approach it gingerly. To see what is

going on, let us look at a very simple polynomial, a monic quadratic (without loss of

generality, all of the polynomials we will consider will be monic) with zeros rand s.

Then, following equation (2), we can write our polynomial as

�+alx+ao = (x - r)(x-s).

The right-hand side is equal to � - rx - sx + rs, and if we equate terms with those on

the left-hand side, we get

al=-(r+s), ao=rs.

Since in general we will need to mUltiply out more complicated expressions, let us

think about how we just did this easy one. We used "FOIL I I," which really just means

"multiply every monomial in (x - r) with every monomial in (x - s)." In other words,

we computed

(x- r)(x-s) = (x+ (-r))(x+ (-s)) =x·x+ (-r) ·x+x· (-s) + (-r) · (-s).

The same procedure works when we multiply out more complicated expressions.

For example, consider

x^3 +a 2 � +alx+ao = (x - q)(x - r)(x-s).

After mUltiplying out the right-hand side, but before collecting like terms, we will have

2·2·2 = 8 terms, since we multiply each monomial in (x - q) with each monomial in

(x - r) with each monomial in (x - s), and each term has just two monomials in it. In

other words, each of the eight terms in (x - q) (x - r) (x - s) represents a three-element

choice, one element chosen from x or -q, one chosen from x or -r and one chosen

from x or -so

So what kind of terms can we get? If our three choices are all x, we end up with the

term x^3. There are three ways that we can choose two xs and one constant, producing

the terms -q�, -rx^2 , -sx^2. Likewise, there are three ways in which we can choose

just one x and two constants, producing qrx, qsx, rsx. Finally, there is just one way

to chose no xs, the term -qrs. This is eight terms in all, and collecting like terms we

have

x^3 +a 2 � +alx+aO = (x - q)(x - r)(x-s)

= x^3 - (q+r+s)� + (qr+qs+rs)x- qrs.

Equating like terms, we see that

a 2 = -(q+r+s), al = qr+qs+rs, ao = -qrs.

Let us do one more example, this time a monic quartic polynomial with zeros

p,q, r, s. We write our polynomial as

x^4 +a 3 x^3 +a 2 � +alx+ao = (x - p)(x-q)(x-r)(x-s).

II This stands for "first, outer, inner, last."