How many and of which kind of roots does the equation have

Question

asked May 9, 2023 174k views

1 Answer

JoGe · Answer 1 · 2023-05-13T07:03:38+0000

The Solution:

Given:

Required:

Find the number and types of roots the function has.

$\begin{gathered} f(1)=1^3-(1^2)-1+1=1-1-1+1=0 \\ This\text{ means that: }x=1\text{ is a root of the function.} \end{gathered}$

Find the other root.

Factorize the function:

$\begin{gathered} f(x)=x^3-x^2-x+1=0 \\ x^2(x-1)-1(x-1)=0 \end{gathered}$
(x-1)(x^2-1)=0

So, the complete roots of the function are:

$\begin{gathered} x-1=0 \\ x=1\text{ \lparen multiplicity of 2\rparen.} \\ x+1=0 \\ x=-1 \end{gathered}$

Thus,

Answer:

The number of roots is 3. (x = 1, x = 1, and x = -1).

The roots are all real roots.

[option D]