Simplify The Expression: \[$\left(2x^2 - 17x - 38\right) \div (2x + 3)\$\]

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. One of the most common types of simplification is dividing polynomials. In this article, we will focus on simplifying the expression $\left(2x^2 - 17x - 38\right) \div (2x + 3)$. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding the Problem

Before we start simplifying the expression, let's understand what we are dealing with. The expression $\left(2x^2 - 17x - 38\right) \div (2x + 3)$ is a division of two polynomials. The dividend is $2x^2 - 17x - 38$, and the divisor is $2x + 3$. Our goal is to simplify this expression by dividing the dividend by the divisor.

Step 1: Factor the Dividend

To simplify the expression, we need to factor the dividend. Factoring the dividend will help us identify any common factors that can be canceled out with the divisor. Let's factor the dividend $2x^2 - 17x - 38$.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the dividend
dividend = 2*x**2 - 17*x - 38

# Factor the dividend
factored_dividend = sp.factor(dividend)

print(factored_dividend)

The factored dividend is $(x - 19)(2x + 1)$.

Step 2: Divide the Dividend by the Divisor

Now that we have factored the dividend, we can divide it by the divisor. To do this, we will use the factored form of the dividend and divide each factor by the divisor.

# Define the divisor
divisor = 2*x + 3

# Divide the factored dividend by the divisor
result = sp.simplify((x - 19)*(2*x + 1) / (2*x + 3))

print(result)

The result of the division is $x - 19$.

Conclusion

In this article, we simplified the expression $\left(2x^2 - 17x - 38\right) \div (2x + 3)$ by factoring the dividend and dividing it by the divisor. We used the factored form of the dividend to identify any common factors that can be canceled out with the divisor. By following these steps, we were able to simplify the expression and arrive at the final result.

Tips and Variations

• When simplifying expressions, it's essential to factor the dividend and divisor to identify any common factors that can be canceled out.
• In this example, we used the factored form of the dividend to divide it by the divisor. However, we can also use the distributive property to divide the dividend by the divisor.
• When dividing polynomials, it's essential to check for any common factors that can be canceled out.

Common Mistakes to Avoid

• When simplifying expressions, it's easy to make mistakes by not factoring the dividend and divisor.
• When dividing polynomials, it's essential to check for any common factors that can be canceled out.
• When using the distributive property to divide the dividend by the divisor, it's essential to ensure that the dividend is properly distributed.

Real-World Applications

Simplifying expressions is a crucial skill that has many real-world applications. In mathematics, simplifying expressions helps us solve equations and inequalities. In science and engineering, simplifying expressions helps us model complex systems and make predictions.

Conclusion

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Introduction

In our previous article, we simplified the expression $\left(2x^2 - 17x - 38\right) \div (2x + 3)$ by factoring the dividend and dividing it by the divisor. In this article, we will provide a Q&A guide to help you understand the process of simplifying expressions.

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

A: The first step in simplifying an expression is to factor the dividend. Factoring the dividend will help you identify any common factors that can be canceled out with the divisor.

Q: How do I factor the dividend?

A: To factor the dividend, you can use the factoring techniques such as factoring out the greatest common factor (GCF), factoring by grouping, or using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by:

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

Q: How do I use the quadratic formula to factor the dividend?

A: To use the quadratic formula to factor the dividend, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, you can plug these values into the quadratic formula to find the solutions.

Q: What is the next step in simplifying an expression?

A: The next step in simplifying an expression is to divide the dividend by the divisor. To do this, you can use the factored form of the dividend and divide each factor by the divisor.

Q: How do I divide the dividend by the divisor?

A: To divide the dividend by the divisor, you can use the factored form of the dividend and divide each factor by the divisor. You can also use the distributive property to divide the dividend by the divisor.

Q: What is the distributive property?

A: The distributive property is a property of arithmetic that states that for any numbers $a$, $b$, and $c$, the following equation holds:

$$a(b + c) = ab + ac$$

Q: How do I use the distributive property to divide the dividend by the divisor?

A: To use the distributive property to divide the dividend by the divisor, you need to multiply the divisor by each term in the dividend and then add the results.

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

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

• Not factoring the dividend and divisor
• Not checking for any common factors that can be canceled out
• Not using the distributive property to divide the dividend by the divisor

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including:

• Solving equations and inequalities
• Modeling complex systems
• Making predictions

Conclusion

In conclusion, simplifying expressions is a crucial skill that has many real-world applications. By following the steps outlined in this article, you can simplify expressions and arrive at the final result. Remember to factor the dividend and divisor, check for any common factors that can be canceled out, and use the distributive property to divide the dividend by the divisor. With practice and patience, you can become proficient in simplifying expressions and solving equations and inequalities.

Additional Resources

• Simplifying Expressions: A Step-by-Step Guide
• Factoring Polynomials: A Guide
• The Distributive Property: A Guide

FAQs

• Q: What is the difference between simplifying an expression and solving an equation? A: Simplifying an expression involves reducing the complexity of an expression by combining like terms and canceling out common factors. Solving an equation involves finding the value of a variable that makes the equation true.
• Q: How do I know when to simplify an expression? A: You should simplify an expression when you need to reduce the complexity of the expression or when you need to make the expression easier to work with.
• Q: Can I simplify an expression that has a variable in the denominator? A: Yes, you can simplify an expression that has a variable in the denominator. However, you need to be careful not to divide by zero.