Multiply And Simplify:${ \frac{24x^2 + 44x - 28}{2x^2 - 7x + 3} \cdot \frac{x^2 + 2x - 15}{9x^2 - 49} }$

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Introduction

In algebra, multiplying and simplifying expressions is a crucial skill that helps us solve complex problems and equations. When we multiply two or more algebraic expressions, we need to follow a set of rules to simplify the resulting expression. In this article, we will explore the process of multiplying and simplifying algebraic expressions, using the given expression as an example.

Understanding the Expression

The given expression is a product of two rational expressions:

{ \frac{24x^2 + 44x - 28}{2x^2 - 7x + 3} \cdot \frac{x^2 + 2x - 15}{9x^2 - 49} \}

To simplify this expression, we need to multiply the numerators and denominators separately.

Multiplying the Numerators

The first step in multiplying the numerators is to distribute each term in the first numerator to each term in the second numerator. This will result in a new expression with multiple terms.

{ (24x^2 + 44x - 28) \cdot (x^2 + 2x - 15) = 24x^4 + 56x^3 - 360x^2 + 88x^3 + 176x^2 - 420x - 28x^2 - 56x + 420 \}

Simplifying the Numerators

Now that we have the expanded expression, we can simplify it by combining like terms. This involves adding or subtracting terms with the same variable and exponent.

{ 24x^4 + 56x^3 - 360x^2 + 88x^3 + 176x^2 - 420x - 28x^2 - 56x + 420 = 24x^4 + 144x^3 - 212x^2 - 476x + 420 \}

Multiplying the Denominators

The next step is to multiply the denominators. This involves multiplying each term in the first denominator to each term in the second denominator.

{ (2x^2 - 7x + 3) \cdot (9x^2 - 49) = 18x^4 - 98x^2 + 27x^2 - 147x + 27x^2 - 147 \}

Simplifying the Denominators

Now that we have the expanded expression, we can simplify it by combining like terms.

{ 18x^4 - 98x^2 + 27x^2 - 147x + 27x^2 - 147 = 18x^4 - 44x^2 - 147x - 147 \}

Multiplying the Numerators and Denominators

Now that we have simplified the numerators and denominators, we can multiply them together.

{ (24x^4 + 144x^3 - 212x^2 - 476x + 420) \cdot (18x^4 - 44x^2 - 147x - 147) \}

Simplifying the Resulting Expression

To simplify the resulting expression, we need to multiply each term in the first expression to each term in the second expression.

{ (24x^4 + 144x^3 - 212x^2 - 476x + 420) \cdot (18x^4 - 44x^2 - 147x - 147) = 432x^8 + 2592x^7 - 3824x^6 - 10944x^5 + 15120x^4 - 8820x^3 - 7056x^2 - 69648x - 55440 \}

Final Simplification

The final step is to simplify the resulting expression by combining like terms.

{ 432x^8 + 2592x^7 - 3824x^6 - 10944x^5 + 15120x^4 - 8820x^3 - 7056x^2 - 69648x - 55440 = 432x^8 + 2592x^7 - 3824x^6 - 10944x^5 + 15120x^4 - 8820x^3 - 7056x^2 - 69648x - 55440 \}

Conclusion

In this article, we have explored the process of multiplying and simplifying algebraic expressions. We have used the given expression as an example and have shown how to multiply the numerators and denominators separately, and then simplify the resulting expression by combining like terms. This process is essential in algebra and is used to solve complex problems and equations.

Tips and Tricks

  • When multiplying and simplifying algebraic expressions, it is essential to follow the order of operations (PEMDAS).
  • When multiplying the numerators and denominators, it is essential to distribute each term in the first expression to each term in the second expression.
  • When simplifying the resulting expression, it is essential to combine like terms.

Common Mistakes

  • Failing to follow the order of operations (PEMDAS).
  • Failing to distribute each term in the first expression to each term in the second expression.
  • Failing to combine like terms.

Real-World Applications

  • Multiplying and simplifying algebraic expressions is used in a variety of real-world applications, including physics, engineering, and computer science.
  • It is used to solve complex problems and equations, and to model real-world phenomena.

Final Thoughts

In conclusion, multiplying and simplifying algebraic expressions is a crucial skill that is used in a variety of real-world applications. By following the steps outlined in this article, you can simplify complex expressions and solve complex problems and equations. Remember to follow the order of operations (PEMDAS), distribute each term in the first expression to each term in the second expression, and combine like terms. With practice and patience, you can become proficient in multiplying and simplifying algebraic expressions.

Introduction

In our previous article, we explored the process of multiplying and simplifying algebraic expressions. In this article, we will answer some of the most frequently asked questions about multiplying and simplifying algebraic expressions.

Q&A

Q: What is the order of operations when multiplying and simplifying algebraic expressions?

A: The order of operations when multiplying and simplifying algebraic expressions is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I distribute each term in the first expression to each term in the second expression?

A: To distribute each term in the first expression to each term in the second expression, you need to multiply each term in the first expression by each term in the second expression. For example, if you have the expression (2x + 3)(x + 4), you would multiply 2x by x and 4, and then multiply 3 by x and 4.

Q: How do I simplify the resulting expression?

A: To simplify the resulting expression, you need to combine like terms. This involves adding or subtracting terms with the same variable and exponent. For example, if you have the expression 2x + 3x, you would combine the two terms to get 5x.

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

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

  • Failing to follow the order of operations (PEMDAS)
  • Failing to distribute each term in the first expression to each term in the second expression
  • Failing to combine like terms
  • Making errors when multiplying and dividing expressions

Q: How do I use multiplying and simplifying algebraic expressions in real-world applications?

A: Multiplying and simplifying algebraic expressions is used in a variety of real-world applications, including physics, engineering, and computer science. It is used to solve complex problems and equations, and to model real-world phenomena.

Q: What are some tips and tricks for multiplying and simplifying algebraic expressions?

A: Some tips and tricks for multiplying and simplifying algebraic expressions include:

  • Using the distributive property to simplify expressions
  • Combining like terms to simplify expressions
  • Using the order of operations (PEMDAS) to simplify expressions
  • Checking your work to ensure that you have not made any errors

Conclusion

In this article, we have answered some of the most frequently asked questions about multiplying and simplifying algebraic expressions. We have covered topics such as the order of operations, distributing terms, simplifying expressions, common mistakes, real-world applications, and tips and tricks. By following the steps outlined in this article, you can become proficient in multiplying and simplifying algebraic expressions.

Final Thoughts

Multiplying and simplifying algebraic expressions is a crucial skill that is used in a variety of real-world applications. By following the steps outlined in this article, you can become proficient in multiplying and simplifying algebraic expressions. Remember to follow the order of operations (PEMDAS), distribute each term in the first expression to each term in the second expression, and combine like terms. With practice and patience, you can become proficient in multiplying and simplifying algebraic expressions.

Additional Resources

  • For more information on multiplying and simplifying algebraic expressions, check out our previous article on the topic.
  • For practice problems and exercises, try using online resources such as Khan Academy or Mathway.
  • For additional tips and tricks, try searching online for "multiplying and simplifying algebraic expressions" or "algebraic expression simplification".

Common Algebraic Expressions

  • Quadratic Expressions: An expression of the form ax^2 + bx + c, where a, b, and c are constants.
  • Polynomial Expressions: An expression of the form a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, and a_0 are constants.
  • Rational Expressions: An expression of the form p(x) / q(x), where p(x) and q(x) are polynomials.

Algebraic Expression Simplification Techniques

  • Distributive Property: A technique used to simplify expressions by distributing each term in the first expression to each term in the second expression.
  • Combining Like Terms: A technique used to simplify expressions by adding or subtracting terms with the same variable and exponent.
  • Order of Operations (PEMDAS): A technique used to simplify expressions by following a specific order of operations (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).