Respuesta :
We are given the following expression:
[tex]\frac{16x^2+8x+1}{(4x+1)^2}[/tex]We are asked to find the restrictions for this expression. The restrictions for a fractional expression is that the denominator must be different to zero, that is mathematically like this:
[tex](4x+1)^2\ne0[/tex]Now we solve for "x", first by taking square root on both sides:
[tex](4x+1)\ne0[/tex]Now we subtract 1 on both sides:
[tex]\begin{gathered} 4x+1-1\ne-1 \\ 4x\ne-1 \end{gathered}[/tex]Now we divide both sides by 4:
[tex]\begin{gathered} \frac{4x}{4}\ne-\frac{1}{4} \\ x\ne-\frac{1}{4} \end{gathered}[/tex]This means that the domain of the expression is restricted to values of "x" different from -1/4. Now we will simplify the expression by factoring the numerator
We factor the numerator using the perfect square trinomial method. We take the square root to the first and third terms of the denominator, and rewrite it like this:
[tex]16x^2+8x+1=(4x+1)^2[/tex]Replacing this in the expression we get:
[tex]\frac{16x^2+8x+1}{(4x+1)^2}=\frac{(4x+1)^2}{(4x+1)^2}=1[/tex]Therefore the expression is equivalent to 1.
