Unknown

1.2. Quadratic Polynomials 9

(c) Let n be a positive integer and let al,... , a,, be n symbols. What is
the coefficient oft’ in the expansion of the product (1 + alt)(l + aat)
(1 + a$)... (1 + ant)? Argue that the coefficient of t’ in the expansion

of (1 + t)” is ( : ) (read: “n choose r”), the number of distinct ways of

choosing r objects from n distinct objects.

1.2 Quadratic Polynomials

Many of the issues which arise for polynomials in general can be illus-
trated in the special case for which the degree is 2. We review results
about quadratics.

Exercises

1. Let p(t) = at2 + bt + c be a quadratic polynomial.

(a) Show that p(t) can be written in the form

• -&(b2 - 4uc).

(b) Use (a) t o d e t ermine all the roots of the equation

t2 - 7t + 12 = 0.

(c) Give a general formula for the roots of a quadratic.

2. (a) Verify that t2 - r2 = (t - r)(t + r).
   (b) Let r be a zero of the polynomial p(t) = at2 + bt + c. Verify that
   p(t) = p(t) -p(r) = (t - r)(at + ar + b).
   (c) Show that r is a zero of a quadratic polynomial p(t) if and only
   if p(t) can be written in the form (t - r)q(t) for some linear
   polynomial q(t).


3. Solve for x the equation

2m(l+ x2) - (1 + m2)(x + m) = 0.

4. Theory of the quadraiic. Let at2 + bt + c be a polynomial whose
   coefficients are complex numbers.

(a) Deduce from Exercise l(a) that u2 + bt + c can be written as a

constant times the square of a linear polynomial if and only if

its discriminant b2 - 4ac vanishes. In this case, show that there

is only one zero of the polynomial.