Answer :
To simplify the given expression \(\frac{b^2+b}{b^3-2b}\), we should start by factoring both the numerator and the denominator where possible.
Step 1: Factor the numerator and the denominator
The numerator is \(b^2 + b\):
[tex]\[ b^2 + b = b(b + 1) \][/tex]
Now, for the denominator \(b^3 - 2b\):
[tex]\[ b^3 - 2b = b(b^2 - 2) \][/tex]
Thus, the expression becomes:
[tex]\[ \frac{b(b + 1)}{b(b^2 - 2)} \][/tex]
Step 2: Cancel out the common factor \(b\)
Since \(b\) is not equal to zero, we can cancel out \(b\) from the numerator and the denominator:
[tex]\[ \frac{b(b + 1)}{b(b^2 - 2)} = \frac{b + 1}{b^2 - 2} \][/tex]
So the simplified form of the given expression is:
[tex]\[ \frac{b+1}{b^2-2} \][/tex]
Step 3: Compare with the provided options
We compare our simplified result with the given choices:
A. \(\frac{b+1}{b^2-2}\)
B. \(b^2\)
C. \(\frac{1}{b-2}\)
D. \(\frac{b}{b^2-2}\)
From the comparison, it is clear that the correct option is:
A. [tex]\(\frac{b+1}{b^2-2}\)[/tex]
Step 1: Factor the numerator and the denominator
The numerator is \(b^2 + b\):
[tex]\[ b^2 + b = b(b + 1) \][/tex]
Now, for the denominator \(b^3 - 2b\):
[tex]\[ b^3 - 2b = b(b^2 - 2) \][/tex]
Thus, the expression becomes:
[tex]\[ \frac{b(b + 1)}{b(b^2 - 2)} \][/tex]
Step 2: Cancel out the common factor \(b\)
Since \(b\) is not equal to zero, we can cancel out \(b\) from the numerator and the denominator:
[tex]\[ \frac{b(b + 1)}{b(b^2 - 2)} = \frac{b + 1}{b^2 - 2} \][/tex]
So the simplified form of the given expression is:
[tex]\[ \frac{b+1}{b^2-2} \][/tex]
Step 3: Compare with the provided options
We compare our simplified result with the given choices:
A. \(\frac{b+1}{b^2-2}\)
B. \(b^2\)
C. \(\frac{1}{b-2}\)
D. \(\frac{b}{b^2-2}\)
From the comparison, it is clear that the correct option is:
A. [tex]\(\frac{b+1}{b^2-2}\)[/tex]
