To find the simplest form of the given expression [tex]\(\frac{4m - 17}{m^2 - 16} + \frac{3m - 11}{m^2 - 16}\)[/tex], we can follow these steps for simplification:
1. Common Denominator: Both fractions already have a common denominator [tex]\(m^2 - 16\)[/tex].
2. Add the Numerators: Since the denominators are the same, we can add the numerators directly:
[tex]\[
\frac{4m - 17}{m^2 - 16} + \frac{3m - 11}{m^2 - 16} = \frac{(4m - 17) + (3m - 11)}{m^2 - 16}
\][/tex]
3. Combine the Numerators:
[tex]\[
\frac{(4m - 17) + (3m - 11)}{m^2 - 16} = \frac{4m + 3m - 17 - 11}{m^2 - 16} = \frac{7m - 28}{m^2 - 16}
\][/tex]
4. Factor the Numerator and Denominator:
The numerator [tex]\(7m - 28\)[/tex] can be factored as [tex]\(7(m - 4)\)[/tex].
[tex]\[
\frac{7m - 28}{m^2 - 16} = \frac{7(m - 4)}{m^2 - 16}
\][/tex]
The denominator [tex]\(m^2 - 16\)[/tex] is a difference of squares, which can be factored as [tex]\((m - 4)(m + 4)\)[/tex].
[tex]\[
m^2 - 16 = (m - 4)(m + 4)
\][/tex]
5. Simplify the Fraction:
[tex]\[
\frac{7(m - 4)}{(m - 4)(m + 4)}
\][/tex]
We can cancel the common factor [tex]\((m - 4)\)[/tex] in the numerator and the denominator (assuming [tex]\(m \neq 4\)[/tex]):
[tex]\[
\frac{7(m - 4)}{(m - 4)(m + 4)} = \frac{7}{m + 4}
\][/tex]
Thus, the simplest form of the expression [tex]\(\frac{4 m - 17}{m^2 - 16} + \frac{3 m - 11}{m^2 - 16}\)[/tex] has [tex]\(7\)[/tex] in the numerator and [tex]\(m + 4\)[/tex] in the denominator.
Therefore, the answer is:
The simplest form of the expression [tex]\(\frac{4m - 17}{m^2 - 16} + \frac{3m - 11}{m^2 - 16}\)[/tex] has [tex]\(7\)[/tex] in the numerator and [tex]\(m + 4\)[/tex] in the denominator.
