Select The Correct Answer.Assuming That No Denominator Equals Zero, What Is The Simplest Form Of This Expression?$\frac{x+2}{x^2+5x+6} \div \frac{3x+1}{z^2-9}$A. $\frac{1}{(x+3)(x-3)}$ B. $\frac{3x+1}{x-3}$ C.

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Introduction

Algebraic expressions can be complex and daunting, especially when dealing with fractions and variables. In this article, we will explore how to simplify a given expression, focusing on the correct order of operations and the rules of algebra. We will use the expression x+2x2+5x+6÷3x+1z2−9\frac{x+2}{x^2+5x+6} \div \frac{3x+1}{z^2-9} as a case study.

Understanding the Expression

The given expression is a division of two fractions. To simplify it, we need to follow the order of operations (PEMDAS):

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

Simplifying the Expression

To simplify the expression, we need to follow the order of operations:

  1. Evaluate the expressions inside the parentheses:
    • x2+5x+6=(x+3)(x+2)x^2+5x+6 = (x+3)(x+2)
    • z2−9=(z+3)(z−3)z^2-9 = (z+3)(z-3)
  2. Rewrite the expression using the simplified expressions:
    • x+2(x+3)(x+2)÷3x+1(z+3)(z−3)\frac{x+2}{(x+3)(x+2)} \div \frac{3x+1}{(z+3)(z-3)}
  3. Cancel out any common factors:
    • 1(x+3)÷3x+1(z+3)(z−3)\frac{1}{(x+3)} \div \frac{3x+1}{(z+3)(z-3)}
  4. Rewrite the expression as a multiplication:
    • 1(x+3)×(z+3)(z−3)3x+1\frac{1}{(x+3)} \times \frac{(z+3)(z-3)}{3x+1}
  5. Simplify the expression:
    • (z+3)(z−3)(x+3)(3x+1)\frac{(z+3)(z-3)}{(x+3)(3x+1)}

Simplifying the Expression Further

We can simplify the expression further by factoring the numerator and denominator:

  • (z+3)(z−3)=z2−9(z+3)(z-3) = z^2-9
  • (x+3)(3x+1)=3x2+10x+3(x+3)(3x+1) = 3x^2+10x+3

The Final Answer

The simplest form of the expression is:

z2−93x2+10x+3\frac{z^2-9}{3x^2+10x+3}

Conclusion

Simplifying complex algebraic expressions requires a step-by-step approach, following the order of operations and the rules of algebra. By breaking down the expression into smaller parts and simplifying each part, we can arrive at the final answer. In this case, the simplest form of the expression is z2−93x2+10x+3\frac{z^2-9}{3x^2+10x+3}.

Discussion

  • What are some common mistakes to avoid when simplifying algebraic expressions?
  • How can we use technology, such as calculators or computer algebra systems, to simplify complex expressions?
  • What are some real-world applications of simplifying algebraic expressions?

Answer Key

A. 1(x+3)(x−3)\frac{1}{(x+3)(x-3)} B. 3x+1x−3\frac{3x+1}{x-3} C. z2−93x2+10x+3\frac{z^2-9}{3x^2+10x+3}

Note: The correct answer is C. z2−93x2+10x+3\frac{z^2-9}{3x^2+10x+3}.

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Introduction

In our previous article, we explored how to simplify a complex algebraic expression. We broke down the expression into smaller parts and simplified each part, following the order of operations and the rules of algebra. In this article, we will answer some frequently asked questions about simplifying complex algebraic expressions.

Q&A

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

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

  • Not following the order of operations (PEMDAS)
  • Not canceling out common factors
  • Not simplifying the numerator and denominator separately
  • Not checking for any errors in the final answer

Q: How can we use technology, such as calculators or computer algebra systems, to simplify complex expressions?

A: Technology can be a powerful tool when simplifying complex algebraic expressions. Calculators and computer algebra systems can help us:

  • Evaluate expressions quickly and accurately
  • Simplify complex expressions with ease
  • Check for errors in the final answer
  • Explore different solutions and scenarios

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

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

  • Physics and engineering: Simplifying complex expressions is essential in physics and engineering, where equations often involve multiple variables and complex operations.
  • Computer science: Simplifying algebraic expressions is crucial in computer science, where algorithms and data structures often involve complex mathematical operations.
  • Economics: Simplifying algebraic expressions is important in economics, where models and equations often involve complex mathematical operations.

Q: How can we simplify expressions with multiple variables?

A: Simplifying expressions with multiple variables requires careful attention to the order of operations and the rules of algebra. Here are some tips:

  • Identify the variables and their relationships
  • Simplify the expression one step at a time
  • Use substitution or elimination methods to simplify the expression
  • Check for any errors in the final answer

Q: How can we simplify expressions with fractions?

A: Simplifying expressions with fractions requires careful attention to the rules of algebra and the order of operations. Here are some tips:

  • Simplify the numerator and denominator separately
  • Cancel out any common factors
  • Use the least common multiple (LCM) to simplify the expression
  • Check for any errors in the final answer

Q: What are some common algebraic identities that can help us simplify expressions?

A: Some common algebraic identities that can help us simplify expressions include:

  • (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2
  • (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2
  • a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)
  • a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Conclusion

Simplifying complex algebraic expressions requires careful attention to the order of operations and the rules of algebra. By following these tips and using technology, we can simplify complex expressions with ease. In this article, we answered some frequently asked questions about simplifying complex algebraic expressions.

Discussion

  • What are some other algebraic identities that can help us simplify expressions?
  • How can we use algebraic identities to simplify complex expressions?
  • What are some real-world applications of simplifying algebraic expressions?

Answer Key

  • Q: What are some common mistakes to avoid when simplifying algebraic expressions? A: Not following the order of operations (PEMDAS), not canceling out common factors, not simplifying the numerator and denominator separately, and not checking for any errors in the final answer.
  • Q: How can we use technology, such as calculators or computer algebra systems, to simplify complex expressions? A: Technology can help us evaluate expressions quickly and accurately, simplify complex expressions with ease, check for errors in the final answer, and explore different solutions and scenarios.
  • Q: What are some real-world applications of simplifying algebraic expressions? A: Simplifying algebraic expressions has many real-world applications, including physics and engineering, computer science, and economics.
  • Q: How can we simplify expressions with multiple variables? A: We can simplify expressions with multiple variables by identifying the variables and their relationships, simplifying the expression one step at a time, using substitution or elimination methods, and checking for any errors in the final answer.
  • Q: How can we simplify expressions with fractions? A: We can simplify expressions with fractions by simplifying the numerator and denominator separately, canceling out any common factors, using the least common multiple (LCM), and checking for any errors in the final answer.
  • Q: What are some common algebraic identities that can help us simplify expressions? A: Some common algebraic identities that can help us simplify expressions include (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2, (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2, a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b), and a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a+b)(a^2 - ab + b^2).