Sure, let's go step-by-step to solve the problem.
The function given is:
[tex]\[ f(x) = x^3 - x \][/tex]
Step 1: Evaluate [tex]\( f(1) \)[/tex]
First, we need to find [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 1^3 - 1 = 1 - 1 = 0 \][/tex]
So, [tex]\( f(1) = 0 \)[/tex].
Step 2: Substitute [tex]\( f(1) \)[/tex] and simplify the expression
Now, we need to compute the expression:
[tex]\[ \frac{f(x) - f(1)}{x - 1} \][/tex]
Substitute [tex]\( f(x) = x^3 - x \)[/tex] and [tex]\( f(1) = 0 \)[/tex]:
[tex]\[ \frac{f(x) - f(1)}{x - 1} = \frac{(x^3 - x) - 0}{x - 1} = \frac{x^3 - x}{x - 1} \][/tex]
Step 3: Simplify the quotient
We now need to simplify the expression [tex]\( \frac{x^3 - x}{x - 1} \)[/tex].
Notice that [tex]\( x^3 - x \)[/tex] can be factored:
[tex]\[ x^3 - x = x(x^2 - 1) \][/tex]
And further factorizing [tex]\( x^2 - 1 \)[/tex]:
[tex]\[ x^2 - 1 = (x - 1)(x + 1) \][/tex]
So:
[tex]\[ x^3 - x = x(x - 1)(x + 1) \][/tex]
Thus, we can write:
[tex]\[ \frac{x^3 - x}{x - 1} = \frac{x(x - 1)(x + 1)}{x - 1} \][/tex]
Since [tex]\( x \neq 1 \)[/tex], we can cancel out [tex]\( x - 1 \)[/tex] in the numerator and denominator:
[tex]\[ \frac{x(x - 1)(x + 1)}{x - 1} = x(x + 1) \][/tex]
So, the simplified form is:
[tex]\[ \frac{x^3 - x}{x - 1} = x(x + 1) \][/tex]
Final Answer
Therefore, the overall expression simplifies to:
[tex]\[ \frac{f(x) - f(1)}{x - 1} = x(x + 1) \][/tex]
