Factor The Expression: 6 X 2 Y − 10 X Y + 15 X − 25 6x^2y - 10xy + 15x - 25 6 X 2 Y − 10 X Y + 15 X − 25

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Introduction

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Factoring an expression is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will focus on factoring the given expression: $6x^2y - 10xy + 15x - 25$. Factoring an expression can help us simplify complex equations, identify common factors, and solve problems more efficiently.

Understanding the Expression

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Before we begin factoring, let's take a closer look at the given expression: $6x^2y - 10xy + 15x - 25$. This expression consists of four terms, each with a different combination of variables and coefficients. To factor this expression, we need to identify any common factors among the terms.

Identifying Common Factors

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One way to identify common factors is to look for the greatest common divisor (GCD) of the coefficients. In this case, the coefficients are 6, -10, 15, and -25. The GCD of these coefficients is 1, which means that there are no common factors among the terms.

However, we can still try to factor out a common factor from each term. Let's start by factoring out the greatest common factor (GCF) of the variables. In this case, the GCF of $x^2y$, $xy$, $x$, and the constant term is $x$. We can factor out $x$ from each term:

$$6x^2y - 10xy + 15x - 25 = x(6xy - 10y + 15 - 25/x)$$

Factoring the Quadratic Expression

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Now that we have factored out $x$ from each term, we are left with a quadratic expression inside the parentheses: $6xy - 10y + 15 - 25/x$. This expression can be factored further using the quadratic formula or by grouping.

Let's try to factor the quadratic expression by grouping. We can group the first two terms and the last two terms:

$$6xy - 10y + 15 - 25/x = (6xy - 10y) + (15 - 25/x)$$

Factoring the Grouped Terms

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Now that we have grouped the terms, we can factor out a common factor from each group. From the first group, we can factor out $2y$:

$$(6xy - 10y) = 2y(3x - 5)$$

From the second group, we can factor out $5$:

$$(15 - 25/x) = 5(3 - 5/x)$$

Combining the Factored Terms

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Now that we have factored the grouped terms, we can combine the factored terms to get the final factored expression:

$$6x^2y - 10xy + 15x - 25 = x(2y(3x - 5) + 5(3 - 5/x))$$

Simplifying the Factored Expression

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The final factored expression can be simplified further by combining like terms:

$$x(2y(3x - 5) + 5(3 - 5/x)) = x(6xy - 10y + 15 - 25/x)$$

Conclusion

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Factoring the expression $6x^2y - 10xy + 15x - 25$ involved identifying common factors, factoring out a common factor from each term, and factoring the quadratic expression inside the parentheses. By following these steps, we were able to simplify the expression and identify its factored form.

Final Answer

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The final factored expression is:

$$x(2y(3x - 5) + 5(3 - 5/x))$$

Tips and Tricks

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• When factoring an expression, always look for common factors among the terms.
• Use the greatest common divisor (GCD) to identify common factors.
• Factor out a common factor from each term to simplify the expression.
• Use the quadratic formula or grouping to factor quadratic expressions.
• Combine like terms to simplify the factored expression.

Common Mistakes

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• Failing to identify common factors among the terms.
• Not factoring out a common factor from each term.
• Not using the quadratic formula or grouping to factor quadratic expressions.
• Not combining like terms to simplify the factored expression.

Real-World Applications

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Factoring expressions has numerous real-world applications in fields such as engineering, physics, and economics. For example, factoring expressions can help us:

• Simplify complex equations and identify common factors.
• Identify patterns and relationships between variables.
• Solve problems more efficiently and accurately.
• Make predictions and forecasts based on data.

Conclusion

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Factoring the expression $6x^2y - 10xy + 15x - 25$ involved identifying common factors, factoring out a common factor from each term, and factoring the quadratic expression inside the parentheses. By following these steps, we were able to simplify the expression and identify its factored form.

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Introduction

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Factoring an expression is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In our previous article, we discussed how to factor the expression $6x^2y - 10xy + 15x - 25$. In this article, we will provide a Q&A guide to help you better understand the concept of factoring and how to apply it to different types of expressions.

Q: What is factoring?

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A: Factoring is the process of expressing a polynomial as a product of simpler polynomials. It involves identifying common factors among the terms and expressing the polynomial as a product of these factors.

Q: Why is factoring important?

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A: Factoring is important because it helps us simplify complex equations and identify common factors. It also helps us solve problems more efficiently and accurately.

Q: How do I factor an expression?

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A: To factor an expression, follow these steps:

1. Identify common factors among the terms.
2. Factor out a common factor from each term.
3. Use the quadratic formula or grouping to factor quadratic expressions.
4. Combine like terms to simplify the factored expression.

Q: What are some common mistakes to avoid when factoring?

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A: Some common mistakes to avoid when factoring include:

• Failing to identify common factors among the terms.
• Not factoring out a common factor from each term.
• Not using the quadratic formula or grouping to factor quadratic expressions.
• Not combining like terms to simplify the factored expression.

Q: How do I factor a quadratic expression?

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A: To factor a quadratic expression, use the quadratic formula or grouping. The quadratic formula is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Grouping involves factoring the quadratic expression into two binomials.

Q: What is the difference between factoring and simplifying?

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A: Factoring involves expressing a polynomial as a product of simpler polynomials, while simplifying involves combining like terms to reduce the complexity of an expression.

Q: Can I factor an expression with a negative coefficient?

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A: Yes, you can factor an expression with a negative coefficient. To do this, simply factor out the negative sign along with the common factor.

Q: How do I factor an expression with a variable in the denominator?

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A: To factor an expression with a variable in the denominator, use the following steps:

1. Factor out the variable from the numerator.
2. Factor out the common factor from the denominator.
3. Combine the factored terms to simplify the expression.

Q: Can I factor an expression with a fraction?

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A: Yes, you can factor an expression with a fraction. To do this, simply factor out the numerator and denominator separately.

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

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A: To factor an expression with a negative exponent, use the following steps:

1. Factor out the variable from the numerator.
2. Factor out the common factor from the denominator.
3. Combine the factored terms to simplify the expression.

Q: Can I factor an expression with a radical?

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A: Yes, you can factor an expression with a radical. To do this, simply factor out the radical along with the common factor.

Conclusion

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Factoring an expression is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. By following the steps outlined in this Q&A guide, you can better understand the concept of factoring and how to apply it to different types of expressions.

Final Tips

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• Practice factoring different types of expressions to become more comfortable with the concept.
• Use the quadratic formula or grouping to factor quadratic expressions.
• Combine like terms to simplify the factored expression.
• Avoid common mistakes such as failing to identify common factors or not factoring out a common factor from each term.

Common Mistakes to Avoid

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• Failing to identify common factors among the terms.
• Not factoring out a common factor from each term.
• Not using the quadratic formula or grouping to factor quadratic expressions.
• Not combining like terms to simplify the factored expression.

Real-World Applications

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Factoring expressions has numerous real-world applications in fields such as engineering, physics, and economics. For example, factoring expressions can help us:

• Simplify complex equations and identify common factors.
• Identify patterns and relationships between variables.
• Solve problems more efficiently and accurately.
• Make predictions and forecasts based on data.

Conclusion

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Factoring the expression $6x^2y - 10xy + 15x - 25$ involved identifying common factors, factoring out a common factor from each term, and factoring the quadratic expression inside the parentheses. By following these steps, we were able to simplify the expression and identify its factored form.