Answer :
To solve the problem of finding the difference and expressing it in simplest form, follow these steps:
1. Identify the common denominator:
The denominators of both fractions are the same: [tex]\(s^2 - 4s + 4\)[/tex].
2. Subtract the numerators:
Since the denominators are the same, we can directly subtract the numerators. The expressions are:
[tex]\[
\frac{8s}{s^2 - 4s + 4} - \frac{16}{s^2 - 4s + 4} = \frac{8s - 16}{s^2 - 4s + 4}
\][/tex]
3. Factor the numerator if possible:
We have the expression [tex]\(8s - 16\)[/tex] in the numerator. We can factor out 8:
[tex]\[
8s - 16 = 8(s - 2)
\][/tex]
4. Examine the denominator:
Let's factor the denominator [tex]\(s^2 - 4s + 4\)[/tex]. This is a perfect square:
[tex]\[
s^2 - 4s + 4 = (s - 2)^2
\][/tex]
5. Simplify the expression:
Substitute the factored forms into the fraction:
[tex]\[
\frac{8(s - 2)}{(s - 2)^2}
\][/tex]
We can cancel [tex]\(s - 2\)[/tex] from the numerator and one [tex]\(s - 2\)[/tex] from the denominator:
[tex]\[
\frac{8}{s - 2}
\][/tex]
Thus, the difference between the two fractions, simplified, is:
[tex]\[
\frac{8}{s - 2}
\][/tex]
This is the simplest form of the expression, resulting in a fully simplified fraction.
1. Identify the common denominator:
The denominators of both fractions are the same: [tex]\(s^2 - 4s + 4\)[/tex].
2. Subtract the numerators:
Since the denominators are the same, we can directly subtract the numerators. The expressions are:
[tex]\[
\frac{8s}{s^2 - 4s + 4} - \frac{16}{s^2 - 4s + 4} = \frac{8s - 16}{s^2 - 4s + 4}
\][/tex]
3. Factor the numerator if possible:
We have the expression [tex]\(8s - 16\)[/tex] in the numerator. We can factor out 8:
[tex]\[
8s - 16 = 8(s - 2)
\][/tex]
4. Examine the denominator:
Let's factor the denominator [tex]\(s^2 - 4s + 4\)[/tex]. This is a perfect square:
[tex]\[
s^2 - 4s + 4 = (s - 2)^2
\][/tex]
5. Simplify the expression:
Substitute the factored forms into the fraction:
[tex]\[
\frac{8(s - 2)}{(s - 2)^2}
\][/tex]
We can cancel [tex]\(s - 2\)[/tex] from the numerator and one [tex]\(s - 2\)[/tex] from the denominator:
[tex]\[
\frac{8}{s - 2}
\][/tex]
Thus, the difference between the two fractions, simplified, is:
[tex]\[
\frac{8}{s - 2}
\][/tex]
This is the simplest form of the expression, resulting in a fully simplified fraction.
