Multiply The Rational Expressions. 50 − 2 W 2 3 W 2 + 9 W − 30 ⋅ W 2 + 5 W − 14 6 W − 30 \frac{50-2w^2}{3w^2+9w-30} \cdot \frac{w^2+5w-14}{6w-30} 3 W 2 + 9 W − 30 50 − 2 W 2 ​ ⋅ 6 W − 30 W 2 + 5 W − 14 ​

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Introduction

Rational expressions are a fundamental concept in algebra, and multiplying them is a crucial skill to master. In this article, we will explore the process of multiplying rational expressions, using the given expression 50−2w23w2+9w−30⋅w2+5w−146w−30\frac{50-2w^2}{3w^2+9w-30} \cdot \frac{w^2+5w-14}{6w-30} as an example. We will break down the steps involved in multiplying rational expressions and provide a clear, step-by-step guide to help you understand the process.

What are Rational Expressions?

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be added, subtracted, multiplied, and divided, just like regular fractions. However, when multiplying rational expressions, we need to follow specific rules to simplify the resulting expression.

The Rules for Multiplying Rational Expressions

When multiplying rational expressions, we need to follow these rules:

  1. Multiply the numerators: Multiply the numerators of the two rational expressions.
  2. Multiply the denominators: Multiply the denominators of the two rational expressions.
  3. Simplify the resulting expression: Simplify the resulting expression by canceling out any common factors in the numerator and denominator.

Multiplying the Given Rational Expressions

Now, let's apply the rules to multiply the given rational expressions:

50−2w23w2+9w−30⋅w2+5w−146w−30\frac{50-2w^2}{3w^2+9w-30} \cdot \frac{w^2+5w-14}{6w-30}

Step 1: Multiply the Numerators

Multiply the numerators of the two rational expressions:

(50−2w2)⋅(w2+5w−14)(50-2w^2) \cdot (w^2+5w-14)

=50w2+250w−700−2w4−10w3+28w2= 50w^2 + 250w - 700 - 2w^4 - 10w^3 + 28w^2

=−2w4−10w3+78w2+250w−700= -2w^4 - 10w^3 + 78w^2 + 250w - 700

Step 2: Multiply the Denominators

Multiply the denominators of the two rational expressions:

(3w2+9w−30)⋅(6w−30)(3w^2+9w-30) \cdot (6w-30)

=18w3−90w2+54w2−270w−180w+900= 18w^3 - 90w^2 + 54w^2 - 270w - 180w + 900

=18w3−36w2−540w+900= 18w^3 - 36w^2 - 540w + 900

Step 3: Simplify the Resulting Expression

Now, we need to simplify the resulting expression by canceling out any common factors in the numerator and denominator.

(−2w4−10w3+78w2+250w−700)÷(18w3−36w2−540w+900)(-2w^4 - 10w^3 + 78w^2 + 250w - 700) \div (18w^3 - 36w^2 - 540w + 900)

To simplify the expression, we need to factor the numerator and denominator.

Factoring the Numerator

Factor the numerator:

−2w4−10w3+78w2+250w−700-2w^4 - 10w^3 + 78w^2 + 250w - 700

=−2w2(w2+5w−350)+250w−700= -2w^2(w^2 + 5w - 350) + 250w - 700

=−2w2(w+25)(w−14)+250w−700= -2w^2(w + 25)(w - 14) + 250w - 700

Factoring the Denominator

Factor the denominator:

18w3−36w2−540w+90018w^3 - 36w^2 - 540w + 900

=18w2(w−2)−540w+900= 18w^2(w - 2) - 540w + 900

=18w2(w−2)−18w(30)+900= 18w^2(w - 2) - 18w(30) + 900

=18w(w−2)(w−5)= 18w(w - 2)(w - 5)

Simplifying the Expression

Now, we can simplify the expression by canceling out any common factors in the numerator and denominator.

−2w2(w+25)(w−14)+250w−70018w(w−2)(w−5)\frac{-2w^2(w + 25)(w - 14) + 250w - 700}{18w(w - 2)(w - 5)}

We can cancel out the common factor ww in the numerator and denominator:

−2w(w+25)(w−14)+250−700w18(w−2)(w−5)\frac{-2w(w + 25)(w - 14) + 250 - \frac{700}{w}}{18(w - 2)(w - 5)}

Conclusion

Multiplying rational expressions requires following specific rules to simplify the resulting expression. By multiplying the numerators and denominators, and then simplifying the resulting expression, we can find the product of two rational expressions. In this article, we used the given expression 50−2w23w2+9w−30⋅w2+5w−146w−30\frac{50-2w^2}{3w^2+9w-30} \cdot \frac{w^2+5w-14}{6w-30} as an example and broke down the steps involved in multiplying rational expressions. We also provided a clear, step-by-step guide to help you understand the process.

Common Mistakes to Avoid

When multiplying rational expressions, it's essential to avoid common mistakes, such as:

  • Not multiplying the numerators and denominators correctly: Make sure to multiply the numerators and denominators separately and then simplify the resulting expression.
  • Not canceling out common factors: Cancel out any common factors in the numerator and denominator to simplify the expression.
  • Not factoring the numerator and denominator: Factor the numerator and denominator to simplify the expression.

By following these rules and avoiding common mistakes, you can master the process of multiplying rational expressions and simplify complex expressions with ease.

Practice Problems

To practice multiplying rational expressions, try the following problems:

  1. x2+4x+4x2−4⋅x2−4x+4x2+4x−4\frac{x^2 + 4x + 4}{x^2 - 4} \cdot \frac{x^2 - 4x + 4}{x^2 + 4x - 4}
  2. 2x2+5x+3x2−3x−4⋅x2+2x−3x2+3x−2\frac{2x^2 + 5x + 3}{x^2 - 3x - 4} \cdot \frac{x^2 + 2x - 3}{x^2 + 3x - 2}
  3. x2−2x−3x2+2x−3⋅x2+3x−2x2−3x−4\frac{x^2 - 2x - 3}{x^2 + 2x - 3} \cdot \frac{x^2 + 3x - 2}{x^2 - 3x - 4}

Conclusion

Introduction

Multiplying rational expressions is a fundamental concept in algebra, and it's essential to understand the process to simplify complex expressions. In this article, we'll provide a Q&A guide to help you master the process of multiplying rational expressions.

Q: What are rational expressions?

A: Rational expressions are fractions that contain variables and/or constants in the numerator and/or denominator.

Q: What are the rules for multiplying rational expressions?

A: The rules for multiplying rational expressions are:

  1. Multiply the numerators: Multiply the numerators of the two rational expressions.
  2. Multiply the denominators: Multiply the denominators of the two rational expressions.
  3. Simplify the resulting expression: Simplify the resulting expression by canceling out any common factors in the numerator and denominator.

Q: How do I multiply the numerators and denominators?

A: To multiply the numerators and denominators, simply multiply the corresponding terms. For example, if you have the expression ab⋅cd\frac{a}{b} \cdot \frac{c}{d}, you would multiply the numerators aa and cc to get acac, and multiply the denominators bb and dd to get bdbd.

Q: How do I simplify the resulting expression?

A: To simplify the resulting expression, you need to cancel out any common factors in the numerator and denominator. This involves factoring the numerator and denominator and canceling out any common factors.

Q: What are some common mistakes to avoid when multiplying rational expressions?

A: Some common mistakes to avoid when multiplying rational expressions include:

  • Not multiplying the numerators and denominators correctly: Make sure to multiply the numerators and denominators separately and then simplify the resulting expression.
  • Not canceling out common factors: Cancel out any common factors in the numerator and denominator to simplify the expression.
  • Not factoring the numerator and denominator: Factor the numerator and denominator to simplify the expression.

Q: How do I factor the numerator and denominator?

A: To factor the numerator and denominator, you need to find the greatest common factor (GCF) of the terms and factor it out. For example, if you have the expression a+bc+d\frac{a+b}{c+d}, you would factor out the GCF of aa and bb to get a+bc+d=a(1+b/a)c(1+d/c)\frac{a+b}{c+d} = \frac{a(1+b/a)}{c(1+d/c)}.

Q: What are some examples of multiplying rational expressions?

A: Here are some examples of multiplying rational expressions:

  • x2+4x+4x2−4⋅x2−4x+4x2+4x−4\frac{x^2 + 4x + 4}{x^2 - 4} \cdot \frac{x^2 - 4x + 4}{x^2 + 4x - 4}
  • 2x2+5x+3x2−3x−4⋅x2+2x−3x2+3x−2\frac{2x^2 + 5x + 3}{x^2 - 3x - 4} \cdot \frac{x^2 + 2x - 3}{x^2 + 3x - 2}
  • x2−2x−3x2+2x−3⋅x2+3x−2x2−3x−4\frac{x^2 - 2x - 3}{x^2 + 2x - 3} \cdot \frac{x^2 + 3x - 2}{x^2 - 3x - 4}

Q: How do I practice multiplying rational expressions?

A: To practice multiplying rational expressions, try the following problems:

  • x2+4x+4x2−4⋅x2−4x+4x2+4x−4\frac{x^2 + 4x + 4}{x^2 - 4} \cdot \frac{x^2 - 4x + 4}{x^2 + 4x - 4}
  • 2x2+5x+3x2−3x−4⋅x2+2x−3x2+3x−2\frac{2x^2 + 5x + 3}{x^2 - 3x - 4} \cdot \frac{x^2 + 2x - 3}{x^2 + 3x - 2}
  • x2−2x−3x2+2x−3⋅x2+3x−2x2−3x−4\frac{x^2 - 2x - 3}{x^2 + 2x - 3} \cdot \frac{x^2 + 3x - 2}{x^2 - 3x - 4}

Conclusion

Multiplying rational expressions is a crucial skill in algebra, and mastering it requires practice and patience. By following the rules and avoiding common mistakes, you can simplify complex expressions with ease. Remember to multiply the numerators and denominators correctly, cancel out common factors, and factor the numerator and denominator to simplify the expression. With practice, you'll become proficient in multiplying rational expressions and tackle complex problems with confidence.

Additional Resources

For more information on multiplying rational expressions, check out the following resources:

  • Khan Academy: Multiplying Rational Expressions
  • Mathway: Multiplying Rational Expressions
  • Wolfram Alpha: Multiplying Rational Expressions

Practice Problems

Try the following problems to practice multiplying rational expressions:

  1. x2+4x+4x2−4⋅x2−4x+4x2+4x−4\frac{x^2 + 4x + 4}{x^2 - 4} \cdot \frac{x^2 - 4x + 4}{x^2 + 4x - 4}
  2. 2x2+5x+3x2−3x−4⋅x2+2x−3x2+3x−2\frac{2x^2 + 5x + 3}{x^2 - 3x - 4} \cdot \frac{x^2 + 2x - 3}{x^2 + 3x - 2}
  3. x2−2x−3x2+2x−3⋅x2+3x−2x2−3x−4\frac{x^2 - 2x - 3}{x^2 + 2x - 3} \cdot \frac{x^2 + 3x - 2}{x^2 - 3x - 4}