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Which of the following is equal to the rational expression when [tex][tex]$x \neq -3$[/tex][/tex]?

[tex]\[ \frac{x^2-9}{x+3} \][/tex]

A. [tex][tex]$x-3$[/tex][/tex]
B. [tex][tex]$\frac{1}{x-3}$[/tex][/tex]
C. [tex][tex]$x+3$[/tex][/tex]
D. [tex][tex]$\frac{x-3}{x+3}$[/tex][/tex]


Sagot :

To determine which of the given options is equal to the rational expression [tex]\(\frac{x^2 - 9}{x + 3}\)[/tex] when [tex]\(x \neq -3\)[/tex], let's simplify the expression step-by-step.

1. Start with the given rational expression:

[tex]\[ \frac{x^2 - 9}{x + 3} \][/tex]

2. Notice that the numerator, [tex]\(x^2 - 9\)[/tex], is a difference of squares. We can factor it as:

[tex]\[ x^2 - 9 = (x + 3)(x - 3) \][/tex]

3. Substitute the factored form of the numerator back into the rational expression:

[tex]\[ \frac{(x + 3)(x - 3)}{x + 3} \][/tex]

4. We can now cancel the common factor [tex]\((x + 3)\)[/tex] in the numerator and the denominator, provided [tex]\(x \neq -3\)[/tex] to avoid division by zero:

[tex]\[ \frac{(x + 3)(x - 3)}{x + 3} = x - 3 \][/tex]

Therefore, the simplified form of the rational expression [tex]\(\frac{x^2 - 9}{x + 3}\)[/tex] when [tex]\(x \neq -3\)[/tex] is:

[tex]\[ x - 3 \][/tex]

Looking at the given options:

A. [tex]\(x - 3\)[/tex]
B. [tex]\(\frac{1}{x - 3}\)[/tex]
C. [tex]\(x + 3\)[/tex]
D. [tex]\(\frac{x - 3}{x + 3}\)[/tex]

The correct answer is:

A. [tex]\(x - 3\)[/tex]