state the non permissible for x^2-12/x+3 and find one equivalnet expression

To find the non-permissible values and an equivalent expression for the expression \(\frac{x^2 - 12}{x + 3}\), follow these steps:

### 1. Finding the Non-Permissible Values
The denominator of the fraction \(x + 3\) must not be zero because division by zero is undefined. Therefore, we need to find values of \(x\) that make the denominator zero:

\[
x + 3 = 0
\]

Solve for \(x\):

\[
x = -3
\]

So, the non-permissible value is \(x = -3\).

### 2. Simplifying the Expression
Now let's simplify the expression \(\frac{x^2 - 12}{x + 3}\). To see if we can simplify it, factor the numerator if possible.

The numerator \(x^2 - 12\) can be rewritten as \(x^2 - 12\), which does not factor nicely using standard algebraic identities (such as the difference of squares or sum/difference of cubes). Therefore, it cannot be factored further with rational numbers.

Since the numerator \(x^2 - 12\) does not factor in a way that cancels with the denominator \(x + 3\), the expression cannot be simplified further.

### 3. Equivalent Expression
Since the expression cannot be simplified, the equivalent expression to \(\frac{x^2 - 12}{x + 3}\) remains the same:

\[
\frac{x^2 - 12}{x + 3}
\]

However, it's crucial to note that this expression is **undefined** at \(x = -3\).

### Summary
- Non-permissible value: \(x = -3\)
- One equivalent expression: \(\frac{x^2 - 12}{x + 3}\) (since it cannot be simplified further)

Would you like further details or have any questions?

### Relative Questions
1. What happens if the numerator can be factored in terms of the denominator?
2. How do you find non-permissible values for more complex rational expressions?
3. What is the importance of identifying non-permissible values in algebra?
4. Can there be more than one non-permissible value in a rational expression?
5. How can rational expressions be simplified when the numerator and denominator share a common factor?

### Tip
Always start by identifying and excluding non-permissible values before simplifying rational expressions to avoid errors in the solution.

Understand how to find non-permissible values and simplify rational expressions such as x^2 - 12 / x + 3. This detailed explanation covers algebraic concepts like non-permissible values, rational expressions, and simplification methods. Suitable for students in Grades 8-10.

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