Question

factor the following polynomial completely into linear factors using real or complex roots. Express how many roots there are. Also, describe the graph of the function. F(x)=x^5-11x^4+13x^3-143x^2+36x-396

Answer 1

An nth degree polynomial has a total of n roots, real or complex (imaginary). Complex roots occur only in conjugate pairs. Therefore if n is even then it is possible to have no real roots, however if n is odd then there has to be at least one real root. For example, since you have given a 5th degree polynomial, there are a total of 5 roots where at least one root is real. Therefore the following cases are possible:

i). 1 real root and 2 pairs of complex conjugate roots
ii). 2 equal and 1 different real roots, 1 pair of complex conjugate roots
iii). 3 different real roots and 1 pair of complex conjugate roots
iv). 2 pairs of equal and real roots, 1 different real root
v). 3 equal real roots and 1 pair of complex and conjugate roots
vi). 3 equal real roots, another two roots that are equal and real
vii). 3 equal and 2 different real roots
viii). 3 different and 2 equal real roots
ix). 5 different real roots

The graph of an odd degree polynomial function extends from quadrant 3 to quadrant 1 if the leading coefficient is positive. If the leading coefficient of a odd degree polynomial function is negative, then the graph extends from quadrant 2 to quadrant 4.