Drag Each Expression To The Correct Location On The Model. Not All Expressions Will Be Used. 5 X 2 + 25 X + 20 7 X \frac{5x^2 + 25x + 20}{7x} 7 X 5 X 2 + 25 X + 20 ​ Determine Where Each Piece Below Belongs To Create A Rational Expression Equivalent To The One Shown Above:-

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Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying rational expressions, focusing on the given expression $\frac{5x^2 + 25x + 20}{7x}$ and determining where each piece belongs to create a rational expression equivalent to the one shown above.

Understanding Rational Expressions

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A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. It is a ratio of two polynomials, where the numerator and denominator are both polynomials. Rational expressions can be simplified by factoring the numerator and denominator, canceling out common factors, and then simplifying the resulting expression.

The Given Expression

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The given expression is $\frac{5x^2 + 25x + 20}{7x}$. To simplify this expression, we need to factor the numerator and denominator, and then cancel out any common factors.

Factoring the Numerator

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The numerator of the given expression is $5x^2 + 25x + 20$. We can factor this expression by finding two numbers whose product is $5 \times 20 = 100$ and whose sum is $25$. These numbers are $10$ and $5$, so we can write the numerator as:

$$5x^2 + 25x + 20 = (5x + 10)(x + 2)$$

Factoring the Denominator

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The denominator of the given expression is $7x$. This expression is already factored, so we can leave it as is.

Canceling Out Common Factors

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Now that we have factored the numerator and denominator, we can cancel out any common factors. In this case, we can cancel out the common factor of $5x$ from the numerator and denominator:

$$\frac{(5x + 10)(x + 2)}{7x} = \frac{(x + 2)}{\frac{7}{5}}$$

Simplifying the Expression

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The expression is now simplified, and we can see that the numerator is $x + 2$ and the denominator is $\frac{7}{5}$. This is the simplified form of the given expression.

Determining Where Each Piece Belongs

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Now that we have simplified the expression, we need to determine where each piece belongs to create a rational expression equivalent to the one shown above. The numerator is $x + 2$, and the denominator is $\frac{7}{5}$. We can create a rational expression equivalent to the one shown above by multiplying the numerator and denominator by the reciprocal of the denominator:

$$\frac{(x + 2)}{\frac{7}{5}} \times \frac{5}{5} = \frac{5(x + 2)}{7}$$

Conclusion

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In this article, we have explored the process of simplifying rational expressions, focusing on the given expression $\frac{5x^2 + 25x + 20}{7x}$. We have factored the numerator and denominator, canceled out common factors, and simplified the resulting expression. We have also determined where each piece belongs to create a rational expression equivalent to the one shown above. By following these steps, we can simplify rational expressions and create equivalent expressions.

Key Takeaways

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• Rational expressions are a fundamental concept in algebra.
• Simplifying rational expressions involves factoring the numerator and denominator, canceling out common factors, and simplifying the resulting expression.
• The given expression $\frac{5x^2 + 25x + 20}{7x}$ can be simplified by factoring the numerator and denominator and canceling out common factors.
• The simplified expression is $\frac{(x + 2)}{\frac{7}{5}}$.
• A rational expression equivalent to the one shown above can be created by multiplying the numerator and denominator by the reciprocal of the denominator.

Practice Problems

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1. Simplify the rational expression $\frac{x^2 + 4x + 4}{x + 2}$.
2. Create a rational expression equivalent to the one shown above by multiplying the numerator and denominator by the reciprocal of the denominator.
3. Simplify the rational expression $\frac{2x^2 + 6x + 4}{x + 1}$.

Answer Key

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1. $\frac{x + 2}{1}$
2. $\frac{(x + 2)}{\frac{1}{1}} \times \frac{1}{1} = \frac{x + 2}{1}$
3. $\frac{2(x + 1)(x + 2)}{x + 1} = 2(x + 2)$
   

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In this article, we will address some of the most common questions related to simplifying rational expressions. Whether you are a student, teacher, or simply looking to improve your understanding of algebra, this article is for you.

Q: What is a rational expression?

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A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. It is a ratio of two polynomials, where the numerator and denominator are both polynomials.

Q: How do I simplify a rational expression?

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A: To simplify a rational expression, you need to follow these steps:

1. Factor the numerator and denominator.
2. Cancel out any common factors.
3. Simplify the resulting expression.

Q: What is factoring?

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A: Factoring is the process of expressing a polynomial as a product of simpler polynomials. For example, the polynomial $x^2 + 4x + 4$ can be factored as $(x + 2)(x + 2)$.

Q: How do I cancel out common factors?

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A: To cancel out common factors, you need to identify the common factors in the numerator and denominator and cancel them out. For example, in the expression $\frac{(x + 2)(x + 2)}{x + 2}$, the common factor $(x + 2)$ can be canceled out.

Q: What is the difference between a rational expression and a rational number?

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A: A rational number is a number that can be expressed as a ratio of two integers, such as $\frac{1}{2}$ or $\frac{3}{4}$. A rational expression, on the other hand, is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: Can I simplify a rational expression with a variable in the denominator?

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A: Yes, you can simplify a rational expression with a variable in the denominator. However, you need to be careful not to divide by zero. For example, the expression $\frac{x}{x}$ can be simplified to $1$, but the expression $\frac{x}{0}$ is undefined.

Q: How do I create a rational expression equivalent to the one shown above?

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A: To create a rational expression equivalent to the one shown above, you need to multiply the numerator and denominator by the reciprocal of the denominator. For example, the expression $\frac{x}{x}$ can be rewritten as $\frac{x}{x} \times \frac{1}{1} = 1$.

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

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

• Dividing by zero
• Canceling out factors that are not common to both the numerator and denominator
• Failing to simplify the resulting expression

Q: How do I check my work when simplifying rational expressions?

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A: To check your work when simplifying rational expressions, you need to follow these steps:

1. Multiply the numerator and denominator by the reciprocal of the denominator.
2. Simplify the resulting expression.
3. Check that the expression is equivalent to the original expression.

Conclusion

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In this article, we have addressed some of the most common questions related to simplifying rational expressions. Whether you are a student, teacher, or simply looking to improve your understanding of algebra, we hope this article has been helpful. Remember to always follow the steps outlined above when simplifying rational expressions, and to check your work carefully to ensure that you have obtained the correct result.

Key Takeaways

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• Rational expressions are a fundamental concept in algebra.
• Simplifying rational expressions involves factoring the numerator and denominator, canceling out common factors, and simplifying the resulting expression.
• The given expression $\frac{5x^2 + 25x + 20}{7x}$ can be simplified by factoring the numerator and denominator and canceling out common factors.
• A rational expression equivalent to the one shown above can be created by multiplying the numerator and denominator by the reciprocal of the denominator.
• Common mistakes to avoid when simplifying rational expressions include dividing by zero, canceling out factors that are not common to both the numerator and denominator, and failing to simplify the resulting expression.

Practice Problems

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1. Simplify the rational expression $\frac{x^2 + 4x + 4}{x + 2}$.
2. Create a rational expression equivalent to the one shown above by multiplying the numerator and denominator by the reciprocal of the denominator.
3. Simplify the rational expression $\frac{2x^2 + 6x + 4}{x + 1}$.

Answer Key

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1. $\frac{x + 2}{1}$
2. $\frac{(x + 2)}{\frac{1}{1}} \times \frac{1}{1} = \frac{x + 2}{1}$
3. $\frac{2(x + 1)(x + 2)}{x + 1} = 2(x + 2)$