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12. Given the function [tex]$f(x) = x^3 - x$[/tex], simplify the following expression:

[tex]\frac{f(x) - f(1)}{x - 1}[/tex]


Sagot :

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]
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