What Is The Product Of The Following Expression?$\[ \frac{2a - 7}{a} \cdot \frac{3a^2}{2a^2 - 11a + 14} \\]A. \[$\frac{3}{a - 2}\$\]B. \[$\frac{3a}{a - 2}\$\]C. \[$\frac{3a}{a + 2}\$\]D. \[$\frac{3}{a + 2}\$\]

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying a given algebraic expression, step by step. We will use the expression 2aβˆ’7aβ‹…3a22a2βˆ’11a+14\frac{2a - 7}{a} \cdot \frac{3a^2}{2a^2 - 11a + 14} as an example and guide you through the simplification process.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at it. The expression consists of two fractions, each with a numerator and a denominator. The first fraction is 2aβˆ’7a\frac{2a - 7}{a}, and the second fraction is 3a22a2βˆ’11a+14\frac{3a^2}{2a^2 - 11a + 14}.

Step 1: Factor the Denominators

To simplify the expression, we need to factor the denominators of both fractions. Let's start with the first fraction. The denominator is aa, which is already factored. Now, let's move on to the second fraction. The denominator is 2a2βˆ’11a+142a^2 - 11a + 14. We can factor this quadratic expression by finding two numbers whose product is 2β‹…14=282 \cdot 14 = 28 and whose sum is βˆ’11-11. The two numbers are βˆ’7-7 and βˆ’4-4, so we can write the denominator as (2aβˆ’7)(aβˆ’2)(2a - 7)(a - 2).

Step 2: Simplify the Expression

Now that we have factored the denominators, we can simplify the expression. We can start by canceling out any common factors between the numerators and denominators. In this case, we can cancel out the factor of aa in the first fraction and the factor of 2aβˆ’72a - 7 in the second fraction.

import sympy as sp

# Define the variables
a = sp.symbols('a')

# Define the expression
expr = (2*a - 7)/a * (3*a**2)/((2*a - 7)*(a - 2))

# Simplify the expression
simplified_expr = sp.simplify(expr)

Step 3: Multiply the Numerators

Now that we have simplified the expression, we can multiply the numerators together. The numerator of the first fraction is 2aβˆ’72a - 7, and the numerator of the second fraction is 3a23a^2. We can multiply these two expressions together to get the final numerator.

# Multiply the numerators
numerator = (2*a - 7) * (3*a**2)

Step 4: Write the Final Expression

Now that we have multiplied the numerators together, we can write the final expression. The final expression is 3a3βˆ’21a2aβˆ’2\frac{3a^3 - 21a^2}{a - 2}.

Conclusion

In this article, we have simplified the algebraic expression 2aβˆ’7aβ‹…3a22a2βˆ’11a+14\frac{2a - 7}{a} \cdot \frac{3a^2}{2a^2 - 11a + 14} step by step. We have factored the denominators, canceled out common factors, multiplied the numerators together, and written the final expression. The final expression is 3a3βˆ’21a2aβˆ’2\frac{3a^3 - 21a^2}{a - 2}.

Answer

The final answer is 3a3βˆ’21a2aβˆ’2\boxed{\frac{3a^3 - 21a^2}{a - 2}}.

However, we need to compare this answer with the given options to determine the correct answer.

Comparing the Answer with the Options

Let's compare the final answer with the given options.

Option A: 3aβˆ’2\frac{3}{a - 2}

Option B: 3aaβˆ’2\frac{3a}{a - 2}

Option C: 3aa+2\frac{3a}{a + 2}

Option D: 3a+2\frac{3}{a + 2}

We can see that the final answer 3a3βˆ’21a2aβˆ’2\frac{3a^3 - 21a^2}{a - 2} is not equal to any of the given options. However, we can simplify the final answer further by factoring out 3a23a^2 from the numerator.

# Factor out 3a^2 from the numerator
numerator = 3*a**2 * (a - 7)

Now, we can see that the final answer is equal to 3a2(aβˆ’7)aβˆ’2\frac{3a^2(a - 7)}{a - 2}, which is equal to 3a(aβˆ’7)1\frac{3a(a - 7)}{1}, which is equal to 3a1\boxed{\frac{3a}{1}}.

However, we can simplify this answer further by canceling out the common factor of aa.

# Cancel out the common factor of a
answer = 3

Now, we can see that the final answer is equal to 3\boxed{3}.

However, we need to compare this answer with the given options to determine the correct answer.

Conclusion

In this article, we have simplified the algebraic expression 2aβˆ’7aβ‹…3a22a2βˆ’11a+14\frac{2a - 7}{a} \cdot \frac{3a^2}{2a^2 - 11a + 14} step by step. We have factored the denominators, canceled out common factors, multiplied the numerators together, and written the final expression. The final expression is 3a3βˆ’21a2aβˆ’2\frac{3a^3 - 21a^2}{a - 2}. We have also compared this answer with the given options and determined that the correct answer is 3aaβˆ’2\boxed{\frac{3a}{a - 2}}.

Final Answer

Introduction

In our previous article, we explored the process of simplifying algebraic expressions step by step. We used the expression 2aβˆ’7aβ‹…3a22a2βˆ’11a+14\frac{2a - 7}{a} \cdot \frac{3a^2}{2a^2 - 11a + 14} as an example and guided you through the simplification process. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

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

A: The first step in simplifying an algebraic expression is to factor the denominators. This involves breaking down the denominator into its prime factors.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the middle term. For example, to factor the quadratic expression 2x2+5x+32x^2 + 5x + 3, you need to find two numbers whose product is 2β‹…3=62 \cdot 3 = 6 and whose sum is 55. The two numbers are 66 and 11, so you can write the quadratic expression as (2x+3)(x+1)(2x + 3)(x + 1).

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

A: Simplifying an algebraic expression involves reducing the expression to its simplest form by canceling out common factors and combining like terms. Solving an equation, on the other hand, involves finding the value of the variable that makes the equation true.

Q: How do I know when to simplify an algebraic expression?

A: You should simplify an algebraic expression whenever you need to reduce it to its simplest form. This is often the case when you are working with complex expressions or when you need to compare two or more expressions.

Q: Can I simplify an algebraic expression by canceling out common factors?

A: Yes, you can simplify an algebraic expression by canceling out common factors. This involves identifying the common factors in the numerator and denominator and canceling them out.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to write the simplified expression in its simplest form. This involves combining like terms and canceling out any remaining common factors.

Q: How do I know if I have simplified an algebraic expression correctly?

A: You can check if you have simplified an algebraic expression correctly by plugging in a value for the variable and checking if the expression evaluates to the correct value.

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

A: Yes, you can use a calculator to simplify an algebraic expression. However, keep in mind that calculators may not always give you the simplest form of the expression.

Conclusion

In this article, we have answered some frequently asked questions about simplifying algebraic expressions. We have covered topics such as factoring denominators, simplifying quadratic expressions, and canceling out common factors. We hope that this article has been helpful in clarifying any confusion you may have had about simplifying algebraic expressions.

Final Tips

  • Always factor the denominators before simplifying an algebraic expression.
  • Use a calculator to check your work and ensure that you have simplified the expression correctly.
  • Simplify algebraic expressions whenever you need to reduce them to their simplest form.
  • Combine like terms and cancel out any remaining common factors to write the simplified expression in its simplest form.

Common Mistakes to Avoid

  • Failing to factor the denominators before simplifying an algebraic expression.
  • Canceling out common factors without checking if they are actually present.
  • Not combining like terms and canceling out any remaining common factors to write the simplified expression in its simplest form.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article and avoiding common mistakes, you can simplify algebraic expressions with confidence. Remember to always factor the denominators, use a calculator to check your work, and simplify algebraic expressions whenever you need to reduce them to their simplest form.