In Example 2, change the middle term of the second factor to 2x.

Data from Example 2

Consider the equation f(x) = (x − 1)³ (x² + x + 1) = 0.
The factor (x − 1)³ shows that there is a triple root of 1, and there is a total of five roots, because the highest-power term would be x⁵ if we were to multiply out the function. To find the other two roots, we use the quadratic formula on the factor (x² + x + 1). This is permissible, because we are finding the values of x for x² + x + 1 = 0.

For this, we have

Thus,

Therefore, the roots of f(x) = 0 are 1, 1, 1,

In the graph of f(x) shown in Fig. 15.1, the zero feature gives a value of x = 1, but the calculator display does not show that it is a triple root, or that there are complex roots. 

Transcribed Image Text:

X = -1± √1-4 2