Question
Simplify the rational expression completely.xx2−16+2x+4−1x−4(1 point)Responses2x−12(x+4)(x−4)the fraction with numerator 2 x minus 12 and denominator open paren x plus 4 close paren times open paren x minus 4 close parenx−6x−4the fraction with numerator x minus 6 and denominator x minus 42x−4the fraction with numerator 2 and denominator x minus 42x−4(x+4)(x−4)
Answers
GPT-4o mini
To simplify the rational expression \(\frac{2x + 4 - 1}{x^2 - 16}\), let's first rewrite the expression clearly:
\[
\frac{2x + 4 - 1}{x^2 - 16} = \frac{2x + 3}{x^2 - 16}
\]
Next, we note that \(x^2 - 16\) is a difference of squares, which can be factored:
\[
x^2 - 16 = (x - 4)(x + 4)
\]
Now we rewrite the expression:
\[
\frac{2x + 3}{(x - 4)(x + 4)}
\]
Next, we need to check if the numerator \(2x + 3\) can be factored to cancel out any common factors with the denominator. However, \(2x + 3\) does not have any linear factors that will cancel with \((x - 4)\) or \((x + 4)\).
Therefore, the simplified expression remains:
\[
\frac{2x + 3}{(x - 4)(x + 4)}
\]
Thus, the completely simplified expression is:
\[
\frac{2x + 3}{(x - 4)(x + 4)}
\]
This can be expressed in response form as:
**The fraction with numerator \(2x + 3\) and denominator \((x + 4)(x - 4)\)**.
If you need it in a full format just remove the explanatory text.
\[
\frac{2x + 4 - 1}{x^2 - 16} = \frac{2x + 3}{x^2 - 16}
\]
Next, we note that \(x^2 - 16\) is a difference of squares, which can be factored:
\[
x^2 - 16 = (x - 4)(x + 4)
\]
Now we rewrite the expression:
\[
\frac{2x + 3}{(x - 4)(x + 4)}
\]
Next, we need to check if the numerator \(2x + 3\) can be factored to cancel out any common factors with the denominator. However, \(2x + 3\) does not have any linear factors that will cancel with \((x - 4)\) or \((x + 4)\).
Therefore, the simplified expression remains:
\[
\frac{2x + 3}{(x - 4)(x + 4)}
\]
Thus, the completely simplified expression is:
\[
\frac{2x + 3}{(x - 4)(x + 4)}
\]
This can be expressed in response form as:
**The fraction with numerator \(2x + 3\) and denominator \((x + 4)(x - 4)\)**.
If you need it in a full format just remove the explanatory text.