Simplify The Expression:$\[ \frac{2x^2 + 10x + 2}{8x - 8} \times \frac{3x - 3}{4x^2 + 20x + 4} \\]

Introduction

Algebraic expressions can be complex and daunting, but with the right techniques and strategies, they can be simplified to reveal their underlying structure. In this article, we will focus on simplifying a given expression involving fractions and polynomials. We will break down the expression into manageable parts, apply various algebraic manipulations, and ultimately arrive at a simplified form.

The Given Expression

The expression we are given is:

$$\frac{2x^2 + 10x + 2}{8x - 8} \times \frac{3x - 3}{4x^2 + 20x + 4}$$

Step 1: Factor the Numerators and Denominators

To simplify the expression, we need to factor the numerators and denominators of both fractions. Let's start with the first fraction:

$$\frac{2x^2 + 10x + 2}{8x - 8}$$

We can factor the numerator as:

$$2x^2 + 10x + 2 = 2(x^2 + 5x + 1)$$

And the denominator can be factored as:

$$8x - 8 = 8(x - 1)$$

So, the first fraction becomes:

$$\frac{2(x^2 + 5x + 1)}{8(x - 1)}$$

Now, let's factor the second fraction:

$$\frac{3x - 3}{4x^2 + 20x + 4}$$

We can factor the numerator as:

$$3x - 3 = 3(x - 1)$$

And the denominator can be factored as:

$$4x^2 + 20x + 4 = 4(x^2 + 5x + 1)$$

So, the second fraction becomes:

$$\frac{3(x - 1)}{4(x^2 + 5x + 1)}$$

Step 2: Cancel Common Factors

Now that we have factored the numerators and denominators, we can cancel common factors between the two fractions. The first fraction has a factor of $2(x^2 + 5x + 1)$ in the numerator and a factor of $8(x - 1)$ in the denominator. The second fraction has a factor of $3(x - 1)$ in the numerator and a factor of $4(x^2 + 5x + 1)$ in the denominator.

We can cancel the common factor of $(x - 1)$ between the two fractions:

$$\frac{2(x^2 + 5x + 1)}{8(x - 1)} \times \frac{3(x - 1)}{4(x^2 + 5x + 1)} = \frac{2 \times 3}{8 \times 4}$$

Step 3: Simplify the Expression

Now that we have canceled the common factors, we can simplify the expression by multiplying the remaining numerators and denominators:

$$\frac{2 \times 3}{8 \times 4} = \frac{6}{32}$$

We can further simplify the expression by dividing both the numerator and denominator by their greatest common divisor, which is 2:

$$\frac{6}{32} = \frac{3}{16}$$

Conclusion

In this article, we simplified a given expression involving fractions and polynomials. We broke down the expression into manageable parts, applied various algebraic manipulations, and ultimately arrived at a simplified form. By factoring the numerators and denominators, canceling common factors, and simplifying the expression, we were able to reveal the underlying structure of the given expression.

Final Answer

The final answer is $\boxed{\frac{3}{16}}$.

Related Topics

• Algebraic manipulation
• Factoring polynomials
• Canceling common factors
• Simplifying expressions

Further Reading

• Algebraic manipulation techniques
• Factoring polynomials strategies
• Canceling common factors examples
• Simplifying expressions exercises
  

Introduction

In our previous article, we simplified a given expression involving fractions and polynomials. We broke down the expression into manageable parts, applied various algebraic manipulations, and ultimately arrived at a simplified form. In this article, we will answer some frequently asked questions related to algebraic manipulation, factoring polynomials, canceling common factors, and simplifying expressions.

Q&A

Q: What is algebraic manipulation?

A: Algebraic manipulation refers to the process of simplifying or rearranging algebraic expressions using various techniques such as factoring, canceling common factors, and combining like terms.

Q: Why is factoring important in algebraic manipulation?

A: Factoring is an essential technique in algebraic manipulation because it allows us to break down complex expressions into simpler parts, making it easier to simplify or solve them.

Q: How do I factor a polynomial?

A: To factor a polynomial, we need to find two or more binomials whose product is equal to the original polynomial. We can use various techniques such as grouping, synthetic division, or the quadratic formula to factor polynomials.

Q: What is canceling common factors?

A: Canceling common factors is a technique used to simplify expressions by canceling out common factors between the numerator and denominator of a fraction.

Q: How do I cancel common factors?

A: To cancel common factors, we need to identify the common factors between the numerator and denominator of a fraction and then cancel them out.

Q: What is the difference between simplifying and solving an expression?

A: Simplifying an expression refers to the process of reducing it to its simplest form using various techniques such as factoring, canceling common factors, and combining like terms. Solving an expression, on the other hand, refers to the process of finding the value of the expression for a given input.

Q: How do I simplify an expression?

A: To simplify an expression, we need to use various techniques such as factoring, canceling common factors, and combining like terms to reduce it to its simplest form.

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

A: Some common mistakes to avoid when simplifying expressions include:

• Not factoring the numerator and denominator
• Not canceling common factors
• Not combining like terms
• Not checking for errors in the simplification process

Q: How do I check my work when simplifying expressions?

A: To check your work when simplifying expressions, you can:

• Plug in values for the variables to see if the expression simplifies to the expected value
• Use a calculator to check the simplified expression
• Compare your simplified expression with the original expression to ensure that it is equivalent

Conclusion

In this article, we answered some frequently asked questions related to algebraic manipulation, factoring polynomials, canceling common factors, and simplifying expressions. We hope that this Q&A guide has provided you with a better understanding of these important concepts and has helped you to improve your skills in simplifying expressions.

Final Tips

• Practice, practice, practice: The more you practice simplifying expressions, the more comfortable you will become with the techniques and the more confident you will be in your ability to simplify expressions.
• Use online resources: There are many online resources available that can help you to learn and practice simplifying expressions, including video tutorials, practice problems, and interactive exercises.
• Seek help when needed: If you are struggling with simplifying expressions, don't be afraid to seek help from a teacher, tutor, or classmate.

Related Topics

• Algebraic manipulation techniques
• Factoring polynomials strategies
• Canceling common factors examples
• Simplifying expressions exercises

Further Reading

• Algebraic manipulation techniques
• Factoring polynomials strategies
• Canceling common factors examples
• Simplifying expressions exercises