Question

Simplify completely quantity 12 x plus 36 over quantity x squared minus 4 x minus 21 and find the restrictions on the variable.

a.) 12 over quantity x minus 7, x ≠ 7
b.) 12 over quantity x minus 7, x ≠ 7, x ≠ −3
c.) quantity x plus 3 over quantity x minus 7, x ≠ 7
d.)quantity x plus 3 over quantity x minus 7, x ≠ 7, x ≠ −3

Answer 1

Option B

12 over quantity x minus 7, x ≠ 7, x ≠ −3

Solution:

Given that we have to simplify quantity 12 x plus 36 over quantity x squared minus 4 x minus 21

So we have to simplify,

`\frac{12x + 36}{ {x}^(2) - 4x - 21 }`

We have to find the restrictions on the variable

The above given Rational function is defined, for

`{x}^(2) - 4x - 21\\e0\\\\(x+3)(x-7)\\e0\\\\x\\e -3,x\\e7`

In order to simplify the above expression, we need to factor both the numerator and the denominator

`\rightarrow \frac{12x + 36}{ {x}^(2) - 4x - 21 } = \frac{12(x + 3)}{ {x}^(2) - 4x - 21}`

For the denominator, we need to split the middle term of the quadratic to get factored form,

`\rightarrow \frac{12x + 36}{ {x}^(2) - 4x - 21 } = \frac{12(x + 3)}{ {x}^(2) - 7x + 3x- 21}`

`\frac{12x + 36}{ {x}^(2) - 4x - 21 } = (12(x + 3))/( x(x - 7) + 3(x- 7))`

Fcatoring the denominator part,

`\frac{12x + 36}{ {x}^(2) - 4x - 21 } = (12(x + 3))/( (x + 3)(x - 7))`

Cancel out common factors to get,

`\frac{12x + 36}{ {x}^(2) - 4x - 21 } = (12)/(x - 7) \\\\where\\\\x\\e -3,x\\e7`

Thus option B is correct. 12 over quantity x minus 7, x ≠ 7, x ≠ −3