Respuesta :

Steps
x^2 - 10x -24/x^2 - 3x - 108
First, factor x^2 - 10x -24: (x + 2)(x - 12)
= (x + 2)(x - 12)/x^2 - 3x - 108
Second, factor x^2 - 3x - 108: (x + 9)(x - 12)
= (x + 2)(x - 12)/(x + 9)(x - 12)
Third, cancel the common factor: x - 12
= x + 2/x + 9

The restrictions on the variable are  [tex](x+9)\neq 0[/tex]  and [tex]x\neq 9[/tex] .

What are restrictions on the variable?

Restrictions on the variable: So, we can never divide by [tex]0[/tex], hence, a restriction on the variable of a rational function with a linear denominator would be any value of the variable that makes the linear denominator equal to [tex]0[/tex].

We have,

[tex]\frac{ (x^2-10x-24)}{(x^2-3x-108)}[/tex]

Now,

Find the factors of Numerator first and then Denominator;

i.e.

Factors of [tex]x^2-10x-24[/tex] ;

[tex]x^2-10x-24[/tex]

Using the middle term split method,

[tex]x^2+2x-12x-24[/tex]

[tex]x(x+2)-12(x+2)[/tex]

[tex](x+2)(x-12)[/tex]

Now,

Factors of [tex]x^2-3x-108[/tex];

[tex]x^2-3x-108[/tex]

[tex]x^2-12x+9x-108[/tex]

[tex]x(x-12)+9(x-12)[/tex]

[tex](x-12)(x+9)[/tex]

Now, rewrite [tex]\frac{ (x^2-10x-24)}{(x^2-3x-108)}[/tex] this in factors form,

i.e.

[tex]\frac{(x+2)(x-12)}{(x-12)(x+9)}[/tex]

Dividing [tex](x-12)[/tex] we get,

⇒ [tex]\frac{(x+2)}{(x+9)}[/tex]

So, this is the simplified form of the given expression.

We know that,

Denominator of any expression can not be zero,

so,

[tex](x+9)\neq 0[/tex]  and [tex]x\neq 9[/tex]

so, these are the restrictions on the variable.

Hence, we can say that the restrictions on the variable are  [tex](x+9)\neq 0[/tex]  and [tex]x\neq 9[/tex] .

To know more about restrictions on the variable click here

https://brainly.com/question/10957504

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