Certainly! Let's find the simplified product step by step for the given expression:
[tex]\[
\frac{(b-5)}{2b} \cdot \frac{b^2 + 3b}{b-5}
\][/tex]
### Step 1: Write the Expression
First, let's clearly write the expression we need to simplify:
[tex]\[
\frac{(b-5)}{2b} \cdot \frac{b^2 + 3b}{b-5}
\][/tex]
### Step 2: Simplify the Numerators and Denominators Individually
Notice that in the product of two fractions, we can multiply the numerators and denominators directly:
[tex]\[
\frac{(b-5)(b^2 + 3b)}{2b(b-5)}
\][/tex]
### Step 3: Cancel Common Factors
Next, we can cancel the common factor of [tex]\((b-5)\)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{\cancel{(b-5)}(b^2 + 3b)}{2b\cancel{(b-5)}}
\][/tex]
This results in:
[tex]\[
\frac{b^2 + 3b}{2b}
\][/tex]
### Step 4: Simplify the Fraction
Now, we simplify the resulting fraction by dividing each term in the numerator by the denominator:
[tex]\[
\frac{b^2}{2b} + \frac{3b}{2b}
\][/tex]
Simplify each term separately:
1. [tex]\(\frac{b^2}{2b} = \frac{b \cdot b}{2b} = \frac{b}{2}\)[/tex]
2. [tex]\(\frac{3b}{2b} = \frac{3 \cdot b}{2 \cdot b} = \frac{3}{2}\)[/tex]
So the combined expression becomes:
[tex]\[
\frac{b}{2} + \frac{3}{2}
\][/tex]
### Step 5: Write the Final Simplified Expression
Combining the terms, we get:
[tex]\[
\frac{b}{2} + \frac{3}{2} = 0.5b + 1.5
\][/tex]
Therefore, the simplified product is:
[tex]\[
0.5b + 1.5
\][/tex]
Comparing with the options provided:
- [tex]\(\frac{b+3}{2}\)[/tex]
- [tex]\(b+3\)[/tex]
- [tex]\(\frac{2}{b+3}\)[/tex]
- [tex]\(2b+6\)[/tex]
The given options do not exactly match [tex]\(0.5b + 1.5\)[/tex], which is the correct simplified form of the product!
