Which Expression Is Equivalent To $\frac{c^2-4}{c+3}+\frac{c+2}{3\left(c^2-9\right)}$?A. $\frac{c+3}{c^2-4}+\frac{c+2}{3\left(c^2-9\right)}$B. $\frac{c^2-4}{c+3} \div \frac{3\left(c^2-9\right)}{c+2}$C. $\frac{c^2-4}{c+3}

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Introduction

In algebra, simplifying complex expressions is a crucial skill that helps in solving various mathematical problems. When dealing with fractions, it's essential to understand the rules of adding and subtracting them. In this article, we will explore the process of simplifying the given expression c2−4c+3+c+23(c2−9)\frac{c^2-4}{c+3}+\frac{c+2}{3\left(c^2-9\right)} and determine which of the provided options is equivalent to it.

Understanding the Expression

The given expression consists of two fractions: c2−4c+3\frac{c^2-4}{c+3} and c+23(c2−9)\frac{c+2}{3\left(c^2-9\right)}. To simplify this expression, we need to find a common denominator for both fractions. The common denominator is the least common multiple (LCM) of the denominators of the two fractions.

Finding the Common Denominator

To find the LCM of c+3c+3 and 3(c2−9)3\left(c^2-9\right), we need to factorize both expressions. The expression c+3c+3 is already in its simplest form. The expression 3(c2−9)3\left(c^2-9\right) can be factorized as 3(c+3)(c−3)3\left(c+3\right)\left(c-3\right).

Simplifying the Expression

Now that we have the common denominator, we can rewrite both fractions with the common denominator. The first fraction becomes c2−4c+3⋅3(c+3)(c−3)3(c+3)(c−3)\frac{c^2-4}{c+3} \cdot \frac{3\left(c+3\right)\left(c-3\right)}{3\left(c+3\right)\left(c-3\right)} and the second fraction becomes c+23(c2−9)⋅c+3c+3\frac{c+2}{3\left(c^2-9\right)} \cdot \frac{c+3}{c+3}.

Combining the Fractions

Now that both fractions have the same denominator, we can combine them by adding the numerators. The expression becomes (c2−4)(3(c+3)(c−3))+(c+2)(c+3)3(c+3)(c−3)\frac{\left(c^2-4\right)\left(3\left(c+3\right)\left(c-3\right)\right) + \left(c+2\right)\left(c+3\right)}{3\left(c+3\right)\left(c-3\right)}.

Simplifying the Numerator

To simplify the numerator, we need to expand and combine like terms. The numerator becomes (c2−4)(3(c+3)(c−3))+(c+2)(c+3)=3(c2−4)(c2−9)+(c+2)(c+3)\left(c^2-4\right)\left(3\left(c+3\right)\left(c-3\right)\right) + \left(c+2\right)\left(c+3\right) = 3\left(c^2-4\right)\left(c^2-9\right) + \left(c+2\right)\left(c+3\right).

Expanding the Numerator

Now that we have the expanded numerator, we can simplify it further. The numerator becomes 3(c2−4)(c2−9)+(c+2)(c+3)=3(c4−13c2+36)+c2+5c+63\left(c^2-4\right)\left(c^2-9\right) + \left(c+2\right)\left(c+3\right) = 3\left(c^4-13c^2+36\right) + c^2+5c+6.

Combining Like Terms

To simplify the numerator further, we need to combine like terms. The numerator becomes 3(c4−13c2+36)+c2+5c+6=3c4−39c2+108+c2+5c+63\left(c^4-13c^2+36\right) + c^2+5c+6 = 3c^4-39c^2+108+c^2+5c+6.

Simplifying the Numerator

Now that we have the combined numerator, we can simplify it further. The numerator becomes 3c4−39c2+108+c2+5c+6=3c4−38c2+5c+1143c^4-39c^2+108+c^2+5c+6 = 3c^4-38c^2+5c+114.

Simplifying the Expression

Now that we have the simplified numerator, we can rewrite the expression as 3c4−38c2+5c+1143(c+3)(c−3)\frac{3c^4-38c^2+5c+114}{3\left(c+3\right)\left(c-3\right)}.

Comparing with the Options

Now that we have the simplified expression, we can compare it with the provided options. The simplified expression is 3c4−38c2+5c+1143(c+3)(c−3)\frac{3c^4-38c^2+5c+114}{3\left(c+3\right)\left(c-3\right)}, which is equivalent to option A.

Conclusion

In conclusion, the expression c2−4c+3+c+23(c2−9)\frac{c^2-4}{c+3}+\frac{c+2}{3\left(c^2-9\right)} is equivalent to option A, which is c+3c2−4+c+23(c2−9)\frac{c+3}{c^2-4}+\frac{c+2}{3\left(c^2-9\right)}. This is because the simplified expression 3c4−38c2+5c+1143(c+3)(c−3)\frac{3c^4-38c^2+5c+114}{3\left(c+3\right)\left(c-3\right)} is equivalent to the expression in option A.

Final Answer

The final answer is option A, which is c+3c2−4+c+23(c2−9)\frac{c+3}{c^2-4}+\frac{c+2}{3\left(c^2-9\right)}.

Introduction

In the previous article, we explored the process of simplifying the expression c2−4c+3+c+23(c2−9)\frac{c^2-4}{c+3}+\frac{c+2}{3\left(c^2-9\right)}. In this article, we will answer some frequently asked questions (FAQs) about simplifying expressions.

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

A: The first step in simplifying an expression is to identify the type of expression and determine the best approach to simplify it. This may involve factoring, canceling out common factors, or using algebraic identities.

Q: How do I simplify a fraction with a variable in the denominator?

A: To simplify a fraction with a variable in the denominator, you need to find the least common multiple (LCM) of the denominator and the numerator. Then, multiply both the numerator and the denominator by the LCM to eliminate the variable in the denominator.

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

A: Simplifying an expression involves reducing the expression to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true. For example, simplifying the expression 2x+32x+3 results in 2x+32x+3, while solving the equation 2x+3=52x+3=5 results in x=1x=1.

Q: Can I simplify an expression with multiple variables?

A: Yes, you can simplify an expression with multiple variables. However, you need to follow the order of operations (PEMDAS) and simplify the expression step by step.

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

A: An expression is simplified when it cannot be reduced further. This means that there are no common factors that can be canceled out, and the expression is in its simplest form.

Q: Can I use a calculator to simplify an expression?

A: Yes, you can use a calculator to simplify an expression. However, it's essential to understand the underlying math and be able to simplify the expression manually.

Q: What are some common algebraic identities that can be used to simplify expressions?

A: Some common algebraic identities that can be used to simplify expressions include:

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

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you need to rewrite the expression with a positive exponent. For example, x−2x^{-2} can be rewritten as 1x2\frac{1}{x^2}.

Q: Can I simplify an expression with a radical?

A: Yes, you can simplify an expression with a radical. However, you need to follow the rules of radicals and simplify the expression step by step.

Conclusion

In conclusion, simplifying expressions is an essential skill in algebra that can be used to solve a wide range of problems. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying expressions and tackle even the most challenging problems.

Final Tips

  • Always follow the order of operations (PEMDAS) when simplifying an expression.
  • Use algebraic identities to simplify expressions.
  • Be careful when simplifying expressions with variables and radicals.
  • Practice regularly to become proficient in simplifying expressions.

Additional Resources

  • Khan Academy: Simplifying Expressions
  • Mathway: Simplifying Expressions
  • Algebra.com: Simplifying Expressions