Question

(5) (a) Let p(2) be a polynomial of the form r3 + ax2 + bx + c. What can you say about p() if you plug in a very, very large value for x? What about plugging in a very large negative number? (b) Give a justification for why p(x) must have a root. Hint: Try to draw it without drawing a root. (C) Show that every 3 x 3 matrix has an eigenvector. (d) Can you generalize your argument to other sizes of matrices?

Answer 1

Answer! Given P(x) = ax²+bx+ c As P(x) is odd degree for very large a very small x value of polynomia depend only higher order term P(n)= lim P(x) = (x²) + vxor O CX->- lim P(x) P(x) is continuous function takes both love value and the (t) vale By Intermediate value property Clyp) must have root ie txost P(xo)=o c) characteristic polynomial of 3x3 is P(x) = ax² + bx²+ cxtd degending on al is i negative as 3 degue polynomial poritive or we con show It take both D, as x 4 x - po by By IXP has noot e eigen value for matrin , elgen value For conepards atbart are eigen Dector positive integen nxn matrix always has eigenvalue. odd -XX