Which Of The Following Is The Product Of The Rational Expressions Shown Below? 2 X + 1 ⋅ 5 3 X \frac{2}{x+1} \cdot \frac{5}{3x} X + 1 2 ​ ⋅ 3 X 5 ​ A. 10 3 X + 3 \frac{10}{3x+3} 3 X + 3 10 ​ B. 5 ( X + 1 ) 6 X \frac{5(x+1)}{6x} 6 X 5 ( X + 1 ) ​ C. 10 3 X 2 + 3 X \frac{10}{3x^2+3x} 3 X 2 + 3 X 10 ​ D. 5 X 2 + 3 \frac{5}{x^2+3} X 2 + 3 5 ​

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Introduction

Rational expressions are a fundamental concept in algebra, and understanding how to multiply them is crucial for solving various mathematical problems. In this article, we will explore the process of multiplying rational expressions, using the given example of 2x+1⋅53x\frac{2}{x+1} \cdot \frac{5}{3x} to illustrate the steps involved.

What are Rational Expressions?

Rational expressions are fractions that contain variables and/or constants in the numerator and/or denominator. They can be written in the form p(x)q(x)\frac{p(x)}{q(x)}, where p(x)p(x) and q(x)q(x) are polynomials. Rational expressions can be simplified, added, subtracted, multiplied, and divided, just like regular fractions.

Multiplying Rational Expressions

To multiply rational expressions, we follow the same rules as multiplying regular fractions. We multiply the numerators together and the denominators together, and then simplify the resulting expression.

Step 1: Multiply the Numerators

The first step in multiplying rational expressions is to multiply the numerators together. In the given example, the numerators are 22 and 55, so we multiply them together to get 2⋅5=102 \cdot 5 = 10.

Step 2: Multiply the Denominators

The next step is to multiply the denominators together. In the given example, the denominators are x+1x+1 and 3x3x, so we multiply them together to get (x+1)⋅3x=3x2+3x(x+1) \cdot 3x = 3x^2 + 3x.

Step 3: Simplify the Resulting Expression

Now that we have multiplied the numerators and denominators together, we can simplify the resulting expression. In this case, we can write the expression as 103x2+3x\frac{10}{3x^2 + 3x}.

Answer

So, the product of the rational expressions 2x+1⋅53x\frac{2}{x+1} \cdot \frac{5}{3x} is 103x2+3x\frac{10}{3x^2 + 3x}.

Conclusion

Multiplying rational expressions is a straightforward process that involves multiplying the numerators and denominators together and then simplifying the resulting expression. By following these steps, we can easily multiply rational expressions and solve various mathematical problems.

Common Mistakes to Avoid

When multiplying rational expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not multiplying the numerators and denominators together: Make sure to multiply the numerators and denominators together, just like you would with regular fractions.
  • Not simplifying the resulting expression: Take the time to simplify the resulting expression, as this will make it easier to work with.
  • Not canceling out common factors: If there are common factors in the numerator and denominator, make sure to cancel them out to simplify the expression.

Real-World Applications

Multiplying rational expressions has many real-world applications, including:

  • Science and Engineering: Rational expressions are used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.
  • Finance: Rational expressions are used to calculate interest rates and investment returns.
  • Computer Science: Rational expressions are used to model algorithms and data structures.

Practice Problems

Here are some practice problems to help you reinforce your understanding of multiplying rational expressions:

  1. Multiply the rational expressions 3x−2⋅4x+1\frac{3}{x-2} \cdot \frac{4}{x+1}.
  2. Multiply the rational expressions 2x2+1⋅3x2−1\frac{2}{x^2 + 1} \cdot \frac{3}{x^2 - 1}.
  3. Multiply the rational expressions 5x+2⋅6x−3\frac{5}{x+2} \cdot \frac{6}{x-3}.

Answer Key

  1. 12x2−x−2\frac{12}{x^2 - x - 2}
  2. 6x4−1\frac{6}{x^4 - 1}
  3. 30x2−x−6\frac{30}{x^2 - x - 6}

Conclusion

Introduction

Multiplying rational expressions is a fundamental concept in algebra that has many real-world applications. In our previous article, we explored the process of multiplying rational expressions and provided a step-by-step guide on how to do it. In this article, we will answer some frequently asked questions about multiplying rational expressions.

Q&A

Q: What is the difference between multiplying rational expressions and multiplying regular fractions?

A: The main difference between multiplying rational expressions and multiplying regular fractions is that rational expressions can have variables in the numerator and/or denominator, whereas regular fractions do not.

Q: How do I multiply rational expressions with different variables in the numerator and denominator?

A: To multiply rational expressions with different variables in the numerator and denominator, you need to multiply the numerators together and the denominators together, just like you would with regular fractions. For example, if you have the rational expressions 2xx+1\frac{2x}{x+1} and 3yy−2\frac{3y}{y-2}, you would multiply the numerators together to get 6xy6xy and the denominators together to get (x+1)(y−2)(x+1)(y-2).

Q: What is the rule for multiplying rational expressions with the same variable in the numerator and denominator?

A: When multiplying rational expressions with the same variable in the numerator and denominator, you can cancel out the common factors. For example, if you have the rational expressions 2xx\frac{2x}{x} and 3xx\frac{3x}{x}, you can cancel out the common factor xx to get 21\frac{2}{1} and 31\frac{3}{1}.

Q: How do I multiply rational expressions with negative exponents?

A: To multiply rational expressions with negative exponents, you need to follow the rules of exponents. For example, if you have the rational expressions 2x2x−3\frac{2x^2}{x^{-3}} and 3x−2x−1\frac{3x^{-2}}{x^{-1}}, you would multiply the numerators together to get 6x46x^4 and the denominators together to get x4x^4.

Q: What is the rule for multiplying rational expressions with zero in the numerator or denominator?

A: When multiplying rational expressions with zero in the numerator or denominator, the result is always zero. For example, if you have the rational expressions 2x\frac{2}{x} and 0x\frac{0}{x}, the result is always zero.

Q: How do I multiply rational expressions with complex numbers in the numerator and/or denominator?

A: To multiply rational expressions with complex numbers in the numerator and/or denominator, you need to follow the rules of complex numbers. For example, if you have the rational expressions 2+3ix+1\frac{2+3i}{x+1} and 3−4ix−2\frac{3-4i}{x-2}, you would multiply the numerators together to get (2+3i)(3−4i)(2+3i)(3-4i) and the denominators together to get (x+1)(x−2)(x+1)(x-2).

Q: What is the rule for multiplying rational expressions with fractions in the numerator and/or denominator?

A: When multiplying rational expressions with fractions in the numerator and/or denominator, you need to multiply the numerators together and the denominators together, just like you would with regular fractions. For example, if you have the rational expressions 23\frac{2}{3} and 34\frac{3}{4}, you would multiply the numerators together to get 66 and the denominators together to get 1212.

Conclusion

Multiplying rational expressions is a fundamental concept in algebra that has many real-world applications. By following the rules outlined in this article, you can easily multiply rational expressions and solve various mathematical problems. Remember to avoid common mistakes, such as not multiplying the numerators and denominators together and not simplifying the resulting expression. With practice, you will become proficient in multiplying rational expressions and be able to apply this skill to a wide range of mathematical problems.

Practice Problems

Here are some practice problems to help you reinforce your understanding of multiplying rational expressions:

  1. Multiply the rational expressions 2xx+1\frac{2x}{x+1} and 3yy−2\frac{3y}{y-2}.
  2. Multiply the rational expressions 2x2x−3\frac{2x^2}{x^{-3}} and 3x−2x−1\frac{3x^{-2}}{x^{-1}}.
  3. Multiply the rational expressions 2x\frac{2}{x} and 0x\frac{0}{x}.
  4. Multiply the rational expressions 2+3ix+1\frac{2+3i}{x+1} and 3−4ix−2\frac{3-4i}{x-2}.
  5. Multiply the rational expressions 23\frac{2}{3} and 34\frac{3}{4}.

Answer Key

  1. 6xy(x+1)(y−2)\frac{6xy}{(x+1)(y-2)}
  2. 6x5x4\frac{6x^5}{x^4}
  3. 00
  4. (2+3i)(3−4i)(x+1)(x−2)\frac{(2+3i)(3-4i)}{(x+1)(x-2)}
  5. 612\frac{6}{12}