Find The Simplified Product. B − 5 2 B ⋅ B 2 + 3 B B − 5 \frac{b-5}{2b} \cdot \frac{b^2+3b}{b-5} 2 B B − 5 ​ ⋅ B − 5 B 2 + 3 B ​ A. B + 3 2 \frac{b+3}{2} 2 B + 3 ​ B. B + 3 B+3 B + 3 C. 2 B + 3 \frac{2}{b+3} B + 3 2 ​ D. 2 B + 6 2b+6 2 B + 6

by ADMIN 242 views

=====================================================

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying a specific algebraic expression, which involves multiplying two fractions. We will break down the process into manageable steps, making it easy to understand and follow along.

The Expression to Simplify

The given expression is:

b−52b⋅b2+3bb−5\frac{b-5}{2b} \cdot \frac{b^2+3b}{b-5}

Our goal is to simplify this expression to its simplest form.

Step 1: Identify Common Factors

The first step in simplifying the expression is to identify any common factors between the two fractions. In this case, we can see that both fractions have a common factor of (b−5)(b-5).

Step 2: Cancel Out Common Factors

Now that we have identified the common factor, we can cancel it out. This means that we will divide both the numerator and the denominator of the first fraction by (b−5)(b-5), and do the same for the second fraction.

import sympy as sp

b = sp.symbols('b')



expr = (b-5)/(2b) * (b**2 + 3b)/(b-5)



simplified_expr = sp.cancel(expr)

Step 3: Simplify the Expression

After canceling out the common factors, we are left with:

b2+3b2b\frac{b^2+3b}{2b}

Now, we can simplify this expression further by factoring out any common factors from the numerator.

Step 4: Factor Out Common Factors

We can see that both terms in the numerator have a common factor of bb. We can factor this out to get:

b(b+3)2b\frac{b(b+3)}{2b}

Step 5: Cancel Out Common Factors Again

Now that we have factored out the common factor, we can cancel it out again. This means that we will divide both the numerator and the denominator by bb.

# Cancel out common factors again
simplified_expr = sp.cancel(simplified_expr)

Step 6: Simplify the Expression Further

After canceling out the common factors again, we are left with:

b+32\frac{b+3}{2}

This is the simplified form of the original expression.

Conclusion

In this article, we have walked through the process of simplifying an algebraic expression step by step. We have identified common factors, canceled them out, and simplified the expression further. The final simplified form of the expression is:

b+32\frac{b+3}{2}

This is the correct answer among the options provided.

Final Answer

The final answer is:

b+32\boxed{\frac{b+3}{2}}

This is the simplified form of the original expression.

=====================================================

Introduction

In our previous article, we walked through the process of simplifying an algebraic expression step by step. However, we understand that sometimes, it's easier to learn through questions and answers. In this article, we will provide a Q&A guide to help you understand the concept of simplifying algebraic expressions.

Q1: What is an algebraic expression?

An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. It is a way to represent a mathematical relationship between variables and constants.

Q2: Why is it important to simplify algebraic expressions?

Simplifying algebraic expressions is important because it helps to:

  • Reduce the complexity of the expression
  • Make it easier to solve
  • Identify patterns and relationships between variables and constants
  • Improve understanding of mathematical concepts

Q3: What are the steps to simplify an algebraic expression?

The steps to simplify an algebraic expression are:

  1. Identify common factors
  2. Cancel out common factors
  3. Simplify the expression further by factoring out common factors
  4. Cancel out common factors again
  5. Simplify the expression further

Q4: How do I identify common factors in an algebraic expression?

Common factors are factors that appear in both the numerator and the denominator of an expression. To identify common factors, look for:

  • Common variables
  • Common constants
  • Common mathematical operations

Q5: What is the difference between canceling out common factors and factoring out common factors?

Canceling out common factors involves dividing both the numerator and the denominator by the common factor. Factoring out common factors involves expressing the numerator as a product of the common factor and another expression.

Q6: How do I know when to cancel out common factors and when to factor out common factors?

Cancel out common factors when:

  • The common factor appears in both the numerator and the denominator
  • The common factor is a variable or a constant

Factor out common factors when:

  • The common factor appears only in the numerator
  • The common factor is a mathematical operation

Q7: What are some common mistakes to avoid when simplifying algebraic expressions?

Some common mistakes to avoid when simplifying algebraic expressions are:

  • Failing to identify common factors
  • Canceling out common factors incorrectly
  • Factoring out common factors incorrectly
  • Not simplifying the expression further

Q8: How do I check my work when simplifying algebraic expressions?

To check your work when simplifying algebraic expressions, follow these steps:

  1. Simplify the expression using the steps outlined above
  2. Check that the simplified expression is equivalent to the original expression
  3. Check that the simplified expression is in its simplest form

Q9: What are some real-world applications of simplifying algebraic expressions?

Simplifying algebraic expressions has many real-world applications, including:

  • Solving systems of equations
  • Modeling real-world phenomena
  • Optimizing functions
  • Analyzing data

Q10: How can I practice simplifying algebraic expressions?

To practice simplifying algebraic expressions, try the following:

  • Work through practice problems
  • Use online resources and tools
  • Join a study group or find a study partner
  • Take online courses or watch video tutorials

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying algebraic expressions. Remember to identify common factors, cancel out common factors, and simplify the expression further to get the correct answer.