Consider the polynomial Qx 4xx 22x 15x2 2x 3 a What is

Consider the polynomial Q(x) = 4x(x - 2)^2(x + 1)^5(x^2 + 2x + 3). (a) What is the degree of this polynomial? (b) Describe the long-run behavior of the the polynomial. (c) List its roots with multiplicities.

Solution

Q(x) = 4x(x -2)^2(x +1)^5(x^2 +2x +3) =

a) Degree of polynomial = x*x^2*x^5*x^2 = x^10   ; 10 is the degree

b) Leading coefficient of Q(x) = 4x^10 So, as x ---> infinity Q(x) ----> infinity

c) Roots = x (multiplicity 1)

x = 2 (multiplicity 2)

x = -1 (multiplicity 5)

x^2 +2x +3 has complex roots as b^2 -4ac <0 ; 4 - 12= -8 <0