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10 1. Fundamentals

(b) Show that if the discriminant of the polynomial does not vanish,

then it has two zeros.

(c) Let m and n denote the zeros of at2 + bt + c. (When the discrimi-

nant vanishes, there is only one zero so in this case we set both m

and n equal to that zero.) Show that at2+bt+c = a(t-m)(t-n).

(d) Show that the sum of the zeros of the quadratic at2 + bt + c is

-b/a, and that the product of the zeros is c/a.

5. For which values of m will the polynomial

m2t2 + 2(m + l)t + 4

have exactly one zero?

6. Let s and p be numbers. Show that the solutions (c, y) of the system

z+y=s

xY=P

are the zeros in some order of the quadratic t2 - st + p.

7. Determine the values of x for which 6x2 - 52 - 4 is negative.


8. Determine those values of k for which the equation

x2+x+2

3x+1

=k

is solvable for real 2.

9. If the domain of the function

x2+2-1

x2 + 32 + 2

is the set of all real numbers, show that it assumes all real values.

10. Given that m and n are the roots of the quadratic 6t2 - 5t - 3, find
    a quadratic whose roots are m - n2 and n - m2, without actually
    finding the values of m and n individually.


11. Let m and n be the roots of the equation t2 + bt + c = 0. Show that
    b and c are the roots of the equation

t2 + (m + n - mn)t - mn(m + n) = 0.

12. (a) Let p(t) and q(t) be two quadratic polynomials with integer co-
    efficients. Prove that, if they have a nonrational zero in common,
    then one must be a constant multiple of the other.