Consider the following f(x) = x³ - 5x² - 25x + 50

Question

(A) Find all real zeros for the polynomial function.

(B) Determine the multiplicity of each zero.

(C) Determine the maximum number of turning points of the graph of the

function.

Answer 1

The Fundamental Theorem of Algebra tells us there are a total of 3 real and complex roots. Using Descartes Rule of Signs, there are 2 sign changes meaning there are either 2 positive real roots or zero positive real roots. Then, using "-x", there is one negative real root because there is only one sign change. Using the Rational Roots Theorem, the possible rational roots, if there are any, are +/- (1, 2, 5, 10, 25, 50) so, as a place to start, let's try -2 by doing synthetic division.

Doesn't work but notice the remainder is +28 (so the function value is +28 at x = -2). Now try -5:

Doesn't work either but since the remainder is -75, we learned that there must be a root between -2 and -5 but, according the the Rational Roots Theorem, there is no rational root in that interval so the root must be irrational. At that point, the best way to proceed forward is to use your graphing calculator to find the roots.

Answer 2

A) To solve this we're gonna factor. first, split up the equation into two equal parts: x³-5x², and -25x+50. look for a common factor in both. in this case it will be (x-5). after factoring that out, you're left with x²(x-5)+25(x-5). you can rewrite this by combining the numbers on the outside as (x-5)(x²+25). now set each factor to 0. x-5=0, and x²+25 =0. x-5=0 will be simplified to x=5. x²+25=0 will turn to x²= -25 to get the x by itself. then, take the square root of both sides. since 25 is negative, it's factors will have i which means they are complex roots. x²= -25 will reduce to x=5i and x=-5i.

B) multiplicity means how many of the same zeros there are. each zero has a multiplicity of one since they each only appear once.

C) the maximum number of turning points is one less than the highest power. in this case, the highest power, or exponent, is 3, so the maximum number of turning points in the graph is 2.

good luck :)