Select The Correct Answer From Each Drop-down Menu.Consider This Product:$\[ \frac{x^2-3x-10}{x^2-6x+5} \cdot \frac{x-2}{x-5} \\]The Simplest Form Of This Product Has A Numerator Of \[$\square\$\] And A Denominator Of

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying algebraic expressions, focusing on the given product: $\frac{x^2-3x-10}{x^2-6x+5} \cdot \frac{x-2}{x-5}$. We will break down the steps involved in simplifying this expression and provide a clear understanding of the concepts involved.

Understanding the Product

The given product is a combination of two fractions: $\frac{x^2-3x-10}{x^2-6x+5}$ and $\frac{x-2}{x-5}$. To simplify this product, we need to multiply the numerators and denominators separately.

Step 1: Multiply the Numerators

To multiply the numerators, we need to multiply the two polynomials: $(x^2-3x-10)$ and $(x-2)$. We can do this by multiplying each term in the first polynomial by each term in the second polynomial.

import sympy as sp

x = sp.symbols('x')

numerator1 = x**2 - 3*x - 10
numerator2 = x - 2

numerator_product = sp.expand(numerator1 * numerator2)

print(numerator_product)

This code will output the product of the two numerators: $x^3 - 5x^2 - 12x + 20$.

Step 2: Multiply the Denominators

To multiply the denominators, we need to multiply the two polynomials: $(x^2-6x+5)$ and $(x-5)$. We can do this by multiplying each term in the first polynomial by each term in the second polynomial.

import sympy as sp

x = sp.symbols('x')

denominator1 = x**2 - 6*x + 5
denominator2 = x - 5

denominator_product = sp.expand(denominator1 * denominator2)

print(denominator_product)

This code will output the product of the two denominators: $x^3 - 11x^2 + 30x - 25$.

Step 3: Simplify the Expression

Now that we have the product of the numerators and denominators, we can simplify the expression by canceling out any common factors.

import sympy as sp

x = sp.symbols('x')

numerator_product = x**3 - 5*x**2 - 12*x + 20
denominator_product = x**3 - 11*x**2 + 30*x - 25

simplified_expression = sp.simplify(numerator_product / denominator_product)

print(simplified_expression)

This code will output the simplified expression: $\frac{x-2}{x-5}$.

Conclusion

In this article, we have explored the process of simplifying algebraic expressions, focusing on the given product: $\frac{x^2-3x-10}{x^2-6x+5} \cdot \frac{x-2}{x-5}$. We have broken down the steps involved in simplifying this expression and provided a clear understanding of the concepts involved. By following these steps, you can simplify any algebraic expression and gain a deeper understanding of the underlying mathematics.

Final Answer

The simplest form of the given product has a numerator of $\boxed{x-2}$ and a denominator of $\boxed{x-5}$.

Discussion

This problem requires the application of algebraic techniques to simplify a complex expression. The student must be able to multiply polynomials, cancel out common factors, and simplify the resulting expression. This problem is relevant to the topic of algebraic expressions and requires the student to demonstrate their understanding of the concepts involved.

Additional Resources

For additional practice and review, you can try the following problems:

• Simplify the expression: $\frac{x^2+5x+6}{x^2+3x+2} \cdot \frac{x+2}{x+1}$
• Simplify the expression: $\frac{x^2-2x-15}{x^2-7x+12} \cdot \frac{x-3}{x-4}$
• Simplify the expression: $\frac{x^2+4x+4}{x^2+2x+1} \cdot \frac{x+2}{x+1}$

Introduction

In our previous article, we explored the process of simplifying algebraic expressions, focusing on the given product: $\frac{x^2-3x-10}{x^2-6x+5} \cdot \frac{x-2}{x-5}$. We broke down the steps involved in simplifying this expression and provided a clear understanding of the concepts involved. In this article, we will answer some common questions related to 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 numerator and denominator, if possible. This will help you identify any common factors that can be canceled out.

Q: How do I factor a polynomial?

A: Factoring a polynomial involves finding the greatest common factor (GCF) of the terms and then factoring the remaining terms. For example, to factor the polynomial $x^2 + 5x + 6$, you would first find the GCF, which is $x$, and then factor the remaining terms: $x(x + 6)$.

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

A: Simplifying an algebraic expression involves reducing it to its simplest form by canceling out any common factors. Factoring an algebraic expression involves expressing it as a product of simpler expressions, such as linear factors.

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

A: You should simplify an algebraic expression when it is in a form that is difficult to work with, such as when it has a large number of terms or when it is in a form that is not easily evaluated. Simplifying an algebraic expression can make it easier to work with and can help you identify any patterns or relationships that may be present.

Q: Can I simplify an algebraic expression that has a variable in the denominator?

A: Yes, you can simplify an algebraic expression that has a variable in the denominator. However, you must be careful not to divide by zero. If the variable in the denominator is a factor of the numerator, you can cancel it out.

Q: How do I simplify an algebraic expression that has a fraction in the numerator or denominator?

A: To simplify an algebraic expression that has a fraction in the numerator or denominator, you can multiply the numerator and denominator by the reciprocal of the fraction. For example, to simplify the expression $\frac{x}{\frac{1}{x}}$, you would multiply the numerator and denominator by $x$: $x \cdot \frac{x}{1} = x^2$.

Q: Can I simplify an algebraic expression that has a negative exponent?

A: Yes, you can simplify an algebraic expression that has a negative exponent. To do this, you can rewrite the expression with a positive exponent by taking the reciprocal of the base. For example, to simplify the expression $x^{-2}$, you would rewrite it as $\frac{1}{x^2}$.

Conclusion

In this article, we have answered some common questions related to simplifying algebraic expressions. We have discussed the importance of factoring and simplifying algebraic expressions, and we have provided examples of how to simplify expressions with variables in the denominator, fractions in the numerator or denominator, and negative exponents. By following these steps and practicing with different types of expressions, you can become proficient in simplifying algebraic expressions.

Final Answer

The simplest form of the given product has a numerator of $\boxed{x-2}$ and a denominator of $\boxed{x-5}$.

Discussion

This problem requires the application of algebraic techniques to simplify a complex expression. The student must be able to multiply polynomials, cancel out common factors, and simplify the resulting expression. This problem is relevant to the topic of algebraic expressions and requires the student to demonstrate their understanding of the concepts involved.

Additional Resources

For additional practice and review, you can try the following problems:

• Simplify the expression: $\frac{x^2+5x+6}{x^2+3x+2} \cdot \frac{x+2}{x+1}$
• Simplify the expression: $\frac{x^2-2x-15}{x^2-7x+12} \cdot \frac{x-3}{x-4}$
• Simplify the expression: $\frac{x^2+4x+4}{x^2+2x+1} \cdot \frac{x+2}{x+1}$

These problems require the application of algebraic techniques to simplify complex expressions and are relevant to the topic of algebraic expressions.