Question

Perform the indicated operation to determine if the givensimplification is correct. If it is correct, select TRUE. If it is notcorrect, select FALSE.(x+7)/(x² + 12x + 35) : (x+3)/(x+5) = 1/(x+3)

Answer 1

In order to simplify this expression, first let's put the denominator of the first fraction in the factored form:

$ax^2+bx+c=a(x-x_1)(x-x_2)$

To do so, let's find the zeros of the polynomial using the quadratic formula:

$\begin{gathered} x^2+12x+35=0 \\ a=1,b=12,c=35 \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x_1=\frac{-12+\sqrt[]{144-140}}{2}=\frac{-12+2}{2}=-5 \\ x_2=\frac{-12-2}{2}=-7 \end{gathered}$

So we have:

$x^2+12x+35=(x+5)(x+7)$

Now, let's simplify the expression by inverting the division (turning it into a product) and canceling the like terms:

$\begin{gathered} \frac{(x+7)}{(x+5)(x+7)}\colon\frac{(x+3)}{(x+5)} \\ =\frac{1}{x+5}\cdot\frac{x+5}{x+3} \\ =\frac{1}{x+3} \end{gathered}$

The right side of the expression is correct, therefore the answer is TRUE.