Question

7. Given f(x) = x 3 (x +

5)2( 2 x-7)
a) What is the degree of this polynomial?
b) List the real zero's of f. Give the multiplicity of each zero.
c) Does the graph off touch or cross the x-axis at each zero?
d) Determine the 'end behavior' of this function. Find the monomial power function that the graph off
resembles for very large positive or negative values of x.

Answer 1

a) 6.

b) 0 (multiplicity 3), -5 (multiplicity 2) and 7/2.

c) and (d) - See below.

Explanation:

(a) f(x) = x^3(x + 5)^2 (2x - 7)

The highest degree of x (= x^3*x^2*x) = 3+2+1 = 6.

(b) The zeros are 0 ( from the x^3 = 0) multiplicity 3, from x + 5 = 0 we have -5 (multiplicity 2) and from 2x - 7 =0, 7/2.

(C) The graph just touches the x axis at (-5, 0) and pass through at (0,0) and (7/2, 0).

(d) The graph rises to the left for very large negative values of x and falls to the right at very large values of x.

At the extreme ends this graph resembles the graph of -x^3.