Simplify $\frac{18t - 6t^2}{10t + 50} \cdot \frac{t^2 + 8t + 15}{t^2 - 9}$.

Feb 28, 2025 by ADMIN 76 views

$Simplify $\frac{18t - 6t^2}{10t + 50} \cdot \frac{t^2 + 8t + 15}{t^2 - 9}$$

Introduction to Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves reducing complex expressions to their simplest form, making it easier to solve equations and manipulate variables. In this article, we will focus on simplifying a specific algebraic expression involving fractions and polynomials.

Understanding the Expression

The given expression is $\frac{18t - 6t^2}{10t + 50} \cdot \frac{t^2 + 8t + 15}{t^2 - 9}$ . To simplify this expression, we need to first factorize the numerator and denominator of each fraction, and then cancel out any common factors.

Factoring the Numerator and Denominator

Let's start by factoring the numerator and denominator of each fraction.

Factoring the First Fraction

The numerator of the first fraction is $18t - 6t^2$ , which can be factored as $6t(3 - t)$ . The denominator is $10t + 50$ , which can be factored as $10(t + 5)$ .

Factoring the Second Fraction

The numerator of the second fraction is $t^2 + 8t + 15$ , which can be factored as $(t + 3)(t + 5)$ . The denominator is $t^2 - 9$ , which can be factored as $(t - 3)(t + 3)$ .

Canceling Out Common Factors

Now that we have factored the numerator and denominator of each fraction, we can cancel out any common factors.

Canceling Out Common Factors in the First Fraction

The first fraction has a common factor of $t$ in the numerator and denominator, which can be canceled out. The first fraction simplifies to $\frac{6(3 - t)}{10(t + 5)}$ .

Canceling Out Common Factors in the Second Fraction

The second fraction has a common factor of $(t + 3)$ in the numerator and denominator, which can be canceled out. The second fraction simplifies to $\frac{t + 5}{t - 3}$ .

Multiplying the Fractions

Now that we have simplified each fraction, we can multiply them together.

Multiplying the Fractions

The expression simplifies to $\frac{6(3 - t)}{10(t + 5)} \cdot \frac{t + 5}{t - 3}$ .

Canceling Out Common Factors in the Product

The product has a common factor of $(t + 5)$ in the numerator and denominator, which can be canceled out. The expression simplifies to $\frac{6(3 - t)}{10(t - 3)}$ .

Final Simplification

The expression can be further simplified by canceling out a common factor of $2$ in the numerator and denominator. The final simplified expression is $\frac{3(3 - t)}{5(t - 3)}$ .

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it requires a deep understanding of factoring, canceling out common factors, and multiplying fractions. By following the steps outlined in this article, we can simplify complex expressions and make them easier to solve. The final simplified expression is $\frac{3(3 - t)}{5(t - 3)}$ .

Additional Tips and Tricks

When simplifying algebraic expressions, it's essential to factorize the numerator and denominator of each fraction.
Cancel out any common factors in the numerator and denominator.
Multiply the fractions together and cancel out any common factors in the product.
Simplify the expression by canceling out any common factors in the numerator and denominator.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, simplifying algebraic expressions can help us solve equations of motion and energy. In engineering, simplifying algebraic expressions can help us design and optimize systems. In economics, simplifying algebraic expressions can help us model and analyze economic systems.

Final Thoughts

Introduction

In our previous article, we simplified the algebraic expression $\frac{18t - 6t^2}{10t + 50} \cdot \frac{t^2 + 8t + 15}{t^2 - 9}$ to $\frac{3(3 - t)}{5(t - 3)}$ . In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to factorize the numerator and denominator of each fraction.

Q: How do I factorize a quadratic expression?

A: To factorize a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the middle term. For example, to factorize $x^2 + 5x + 6$ , you need to find two numbers whose product is $6$ and whose sum is $5$ . The numbers are $2$ and $3$ , so the expression can be factored as $(x + 2)(x + 3)$ .

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

A: Factoring involves breaking down an expression into its simplest form by finding the factors of the numerator and denominator. Canceling out common factors involves removing any common factors from the numerator and denominator.

Q: How do I multiply fractions?

A: To multiply fractions, you need to multiply the numerators together and multiply the denominators together. For example, to multiply $\frac{2}{3} \cdot \frac{4}{5}$ , you need to multiply $2$ and $4$ together to get $8$ , and multiply $3$ and $5$ together to get $15$ . The result is $\frac{8}{15}$ .

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to simplify the expression by canceling out any common factors in the numerator and denominator.

Q: Can I simplify an algebraic expression by canceling out common factors without factoring?

A: No, you cannot simplify an algebraic expression by canceling out common factors without factoring. Factoring is a crucial step in simplifying algebraic expressions, as it allows you to identify and cancel out common factors.

Q: How do I know if an algebraic expression is already simplified?

A: An algebraic expression is already simplified if it cannot be simplified further by factoring or canceling out common factors.

Q: Can I simplify an algebraic expression with variables?

A: Yes, you can simplify an algebraic expression with variables. The process is the same as simplifying an algebraic expression with constants.

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

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

Not factoring the numerator and denominator of each fraction
Not canceling out common factors
Not simplifying the expression by canceling out common factors in the numerator and denominator
Not checking if the expression can be simplified further

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of factoring, canceling out common factors, and multiplying fractions. By following the steps outlined in this article, we can simplify complex expressions and make them easier to solve. Remember to factorize the numerator and denominator of each fraction, cancel out any common factors, multiply the fractions together, and simplify the expression by canceling out any common factors in the numerator and denominator.

Additional Tips and Tricks

When simplifying algebraic expressions, it's essential to factorize the numerator and denominator of each fraction.
Cancel out any common factors in the numerator and denominator.
Multiply the fractions together and cancel out any common factors in the product.
Simplify the expression by canceling out any common factors in the numerator and denominator.
Check if the expression can be simplified further by factoring or canceling out common factors.