Fully Simplify The Expression Below And Write Your Answer As A Single Fraction. X 2 + 5 X + 4 X 2 − 8 X ⋅ 6 X 2 − 102 X + 432 4 X 2 − 20 X − 144 \frac{x^2+5x+4}{x^2-8x} \cdot \frac{6x^2-102x+432}{4x^2-20x-144} X 2 − 8 X X 2 + 5 X + 4 ​ ⋅ 4 X 2 − 20 X − 144 6 X 2 − 102 X + 432 ​

Introduction

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves breaking down intricate expressions into simpler forms, making them easier to work with and understand. In this article, we will focus on simplifying the given expression: $\frac{x^2+5x+4}{x^2-8x} \cdot \frac{6x^2-102x+432}{4x^2-20x-144}$. We will break down the process into manageable steps, using various techniques to simplify the expression.

Step 1: Factorize the Numerators and Denominators

To simplify the given expression, we need to factorize the numerators and denominators of both fractions. This will help us identify common factors and cancel them out.

Factorizing the First Numerator

The first numerator is $x^2+5x+4$. We can factorize it as follows:

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

Factorizing the First Denominator

The first denominator is $x^2-8x$. We can factorize it as follows:

$x^2-8x = x(x-8)$

Factorizing the Second Numerator

The second numerator is $6x^2-102x+432$. We can factorize it as follows:

$6x^2-102x+432 = 6(x^2-17x+72) = 6(x-9)(x-8)$

Factorizing the Second Denominator

The second denominator is $4x^2-20x-144$. We can factorize it as follows:

$4x^2-20x-144 = 4(x^2-5x-36) = 4(x-9)(x+4)$

Step 2: Cancel Out Common Factors

Now that we have factorized the numerators and denominators, we can cancel out common factors. The expression becomes:

$\frac{(x+1)(x+4)}{x(x-8)} \cdot \frac{6(x-9)(x-8)}{4(x-9)(x+4)}$

We can cancel out the common factors $(x+4)$ and $(x-8)$ from the numerators and denominators.

Step 3: Simplify the Expression

After canceling out the common factors, the expression becomes:

$\frac{(x+1)}{x} \cdot \frac{6(x-9)}{4}$

We can simplify the expression further by multiplying the numerators and denominators.

Step 4: Simplify the Numerators and Denominators

The expression becomes:

$\frac{6(x+1)(x-9)}{4x}$

We can simplify the numerator by multiplying the factors.

Step 5: Simplify the Final Expression

The final expression becomes:

$\frac{6(x^2-8x+9)}{4x}$

We can simplify the numerator by combining like terms.

Step 6: Simplify the Final Expression

The final expression becomes:

$\frac{6x^2-48x+54}{4x}$

We can simplify the expression further by dividing the numerator and denominator by their greatest common divisor.

Step 7: Simplify the Final Expression

The final expression becomes:

$\frac{3x^2-24x+27}{2x}$

We have successfully simplified the given expression.

Conclusion

Simplifying complex algebraic expressions is a crucial skill in mathematics. By breaking down intricate expressions into simpler forms, we can make them easier to work with and understand. In this article, we have focused on simplifying the given expression: $\frac{x^2+5x+4}{x^2-8x} \cdot \frac{6x^2-102x+432}{4x^2-20x-144}$. We have used various techniques, including factorization and cancellation of common factors, to simplify the expression. The final simplified expression is $\frac{3x^2-24x+27}{2x}$.

Introduction

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. In our previous article, we walked through the step-by-step process of simplifying the expression: $\frac{x^2+5x+4}{x^2-8x} \cdot \frac{6x^2-102x+432}{4x^2-20x-144}$. In this article, we will address some common questions and concerns related to simplifying complex algebraic expressions.

Q: What are the most common techniques used to simplify complex algebraic expressions?

A: The most common techniques used to simplify complex algebraic expressions include:

• Factorization: breaking down an expression into simpler factors
• Cancellation of common factors: canceling out common factors between the numerator and denominator
• Simplification of fractions: simplifying fractions by dividing the numerator and denominator by their greatest common divisor
• Combining like terms: combining like terms in the numerator and denominator

Q: How do I know when to use factorization versus cancellation of common factors?

A: Factorization is typically used when the expression can be broken down into simpler factors, such as quadratic expressions. Cancellation of common factors is typically used when there are common factors between the numerator and denominator.

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

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

• Not factoring the numerator and denominator properly
• Not canceling out common factors correctly
• Not simplifying fractions properly
• Not combining like terms correctly

Q: How do I know when an expression is fully simplified?

A: An expression is fully simplified when there are no more common factors to cancel out, and the numerator and denominator are in their simplest form.

Q: Can I use technology to simplify complex algebraic expressions?

A: Yes, technology can be used to simplify complex algebraic expressions. Many graphing calculators and computer algebra systems (CAS) can simplify expressions automatically.

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

A: To check your work, you can:

• Plug in values for the variables to see if the expression simplifies correctly
• Use a calculator or computer algebra system to check the expression
• Simplify the expression again to see if it matches the original expression

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

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

• Physics: simplifying expressions to model the motion of objects
• Engineering: simplifying expressions to design and optimize systems
• Economics: simplifying expressions to model economic systems

Conclusion

Simplifying complex algebraic expressions is a crucial skill in mathematics, and it has many real-world applications. By understanding the techniques and common mistakes to avoid, you can simplify complex expressions with confidence. Remember to check your work and use technology when necessary to ensure accuracy.