Question

For the​ polynomial,

f(x)=x3−5x2−25x+125
​(a) List each real zero and its multiplicity. Determine whether the graph crosses or touches the ​x-axis at each ​x-intercept.
​(b) Give the power function that the graph of f resembles for large values of​ |x|. Determine which of the following best describes the end​ behavior:
​(i) Rises on the left and rises on the right
​(ii) Rises on the left and falls on the right
​(iii) Falls on the left and rises on the right
​(iv) Falls on the left and falls on the right

Answer 1

The real zeros and multiplicities of the polynomial f(x)=x^3-5x^2-25x+125 are x = -5 (multiplicity 1) and x = 1 (multiplicity 2). The graph of f crosses the x-axis at x = -5 and touches the x-axis at x = 1. For large values of |x|, the graph resembles the power function y = x^3, and the end behavior is falling on the left and rising on the right.

Step-by-step explanation:

a)

The real zeros of the polynomial f(x)=x^3-5x^2-25x+125 can be found by factoring or using the Rational Root Theorem. By testing possible integer roots, it can be determined that the real zeros are x = -5 and x = 1, with multiplicities 1 and 2 respectively.

At the x-intercept x = -5, the graph of f crosses the x-axis, while at the x-intercept x = 1, the graph touches the x-axis but does not cross it.

b)

For large values of |x|, the graph of f(x) will resemble the power function y = x^3, as this is the highest degree term in the polynomial. The end behavior of this power function is falling on the left and rising on the right, which corresponds to option (iii) Falls on the left and rises on the right.