Find the simplified product. [tex]$\frac{b-5}{2 b} \cdot \frac{b^2+3 b}{b-5}$[/tex]
Answer (2)
Factor the numerator of the second fraction: b 2 + 3 b = b ( b + 3 ) .
Rewrite the expression as 2 b b − 5 ⋅ b − 5 b ( b + 3 ) .
Cancel the common factors ( b − 5 ) and b from the numerator and denominator.
The simplified expression is 2 b + 3 .
Explanation
Understanding the Problem We are asked to simplify the product of two rational expressions: 2 b b − 5 ⋅ b − 5 b 2 + 3 b . To do this, we will factor the numerators and denominators of the fractions and then cancel any common factors.
Factoring the Numerator First, we factor the numerator of the second fraction. We have b 2 + 3 b = b ( b + 3 ) . So the expression becomes 2 b b − 5 ⋅ b − 5 b ( b + 3 ) .
Canceling Common Factors Now we can cancel common factors. We have a factor of ( b − 5 ) in both the numerator and denominator, and we also have a factor of b in both the numerator and denominator. Canceling these factors, we get 2 b b − 5 ⋅ b − 5 b ( b + 3 ) = 2 b + 3 .
Final Answer Therefore, the simplified product is 2 b + 3 .
Examples
Rational expressions are useful in many areas, such as physics, engineering, and economics. For example, in physics, they can be used to describe the relationship between the voltage, current, and resistance in an electrical circuit. Simplifying rational expressions can help to make these relationships easier to understand and work with. In this case, simplifying the product of two rational expressions allows us to more easily see the relationship between the variables involved.
To simplify the expression 2 b b − 5 ⋅ b − 5 b 2 + 3 b , we factor the second numerator and cancel common factors. The final simplified product is 2 b + 3 .
;
