Simplify The Expression: ( 2 X 2 + 2 X − 6 ) ( 3 X − 4 \left(2x^2 + 2x - 6\right)(3x - 4 ( 2 X 2 + 2 X − 6 ) ( 3 X − 4 ]

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. One of the most common types of expressions to simplify is the product of two polynomials. In this article, we will focus on simplifying the expression $\left(2x^2 + 2x - 6\right)(3x - 4)$ using the distributive property and combining like terms.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

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

This property can be extended to polynomials, where we can multiply each term in one polynomial by each term in the other polynomial.

Simplifying the Expression

To simplify the expression $\left(2x^2 + 2x - 6\right)(3x - 4)$, we will use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.

Step 1: Multiply Each Term in the First Polynomial by Each Term in the Second Polynomial

We will start by multiplying each term in the first polynomial by the first term in the second polynomial, which is $3x$. Then, we will multiply each term in the first polynomial by the second term in the second polynomial, which is $-4$.

import sympy as sp
x = sp.symbols('x')

poly1 = 2x**2 + 2x - 6
poly2 = 3*x - 4

result = sp.expand(poly1 * poly2)
print(result)

Step 2: Combine Like Terms

After multiplying each term in the first polynomial by each term in the second polynomial, we will combine like terms to simplify the expression.

# Combine like terms
result = sp.simplify(result)
print(result)

The Final Answer

After simplifying the expression using the distributive property and combining like terms, we get:

$$6x^3 - 8x^2 + 6x - 24$$

Conclusion

Simplifying expressions is an essential skill in algebra that helps us solve equations and inequalities. By using the distributive property and combining like terms, we can simplify complex expressions like $\left(2x^2 + 2x - 6\right)(3x - 4)$. In this article, we have shown how to simplify this expression step-by-step using Python code.

Tips and Variations

• To simplify more complex expressions, you can use the sympy library in Python to expand and simplify the expression.
• To check your answer, you can use the sympy library to simplify the expression and compare it with the final answer.
• To practice simplifying expressions, you can try simplifying different types of expressions, such as the product of two binomials or the sum of two polynomials.

Common Mistakes

• Not using the distributive property to multiply each term in one polynomial by each term in the other polynomial.
• Not combining like terms to simplify the expression.
• Not checking the answer using the sympy library.

Real-World Applications

Simplifying expressions is a crucial skill in many real-world applications, such as:

• Physics: Simplifying expressions is essential in physics to solve equations and inequalities that describe the motion of objects.
• Engineering: Simplifying expressions is crucial in engineering to design and optimize systems.
• Computer Science: Simplifying expressions is essential in computer science to optimize algorithms and data structures.

Introduction

In our previous article, we discussed how to simplify the expression $\left(2x^2 + 2x - 6\right)(3x - 4)$ using the distributive property and combining like terms. In this article, we will provide a Q&A guide to help you understand the concept of simplifying expressions and how to apply it in different scenarios.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

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

This property can be extended to polynomials, where we can multiply each term in one polynomial by each term in the other polynomial.

Q: How do I simplify an expression using the distributive property?

A: To simplify an expression using the distributive property, you need to multiply each term in one polynomial by each term in the other polynomial. Then, you need to combine like terms to simplify the expression.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the expression $2x^2 + 3x^2$, you can combine the like terms by adding the coefficients:

$$2x^2 + 3x^2 = 5x^2$$

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

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

• Not using the distributive property to multiply each term in one polynomial by each term in the other polynomial.
• Not combining like terms to simplify the expression.
• Not checking the answer using the sympy library.

Q: How do I check my answer using the sympy library?

A: To check your answer using the sympy library, you can use the sympy library to simplify the expression and compare it with your answer. For example:

import sympy as sp

x = sp.symbols('x')

poly1 = 2x**2 + 2x - 6
poly2 = 3*x - 4

result = sp.expand(poly1 * poly2)

result = sp.simplify(result)

print(result)

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

A: Simplifying expressions is a crucial skill in many real-world applications, such as:

• Physics: Simplifying expressions is essential in physics to solve equations and inequalities that describe the motion of objects.
• Engineering: Simplifying expressions is crucial in engineering to design and optimize systems.
• Computer Science: Simplifying expressions is essential in computer science to optimize algorithms and data structures.

Conclusion

Simplifying expressions is a fundamental concept in algebra that helps us solve equations and inequalities. By mastering the skill of simplifying expressions, you can solve complex problems in various fields and make a significant impact in your career. In this article, we have provided a Q&A guide to help you understand the concept of simplifying expressions and how to apply it in different scenarios.

Tips and Variations

• To simplify more complex expressions, you can use the sympy library in Python to expand and simplify the expression.
• To check your answer, you can use the sympy library to simplify the expression and compare it with your answer.
• To practice simplifying expressions, you can try simplifying different types of expressions, such as the product of two binomials or the sum of two polynomials.

Common Mistakes

• Not using the distributive property to multiply each term in one polynomial by each term in the other polynomial.
• Not combining like terms to simplify the expression.
• Not checking the answer using the sympy library.

Real-World Applications

Simplifying expressions is a crucial skill in many real-world applications, such as:

• Physics: Simplifying expressions is essential in physics to solve equations and inequalities that describe the motion of objects.
• Engineering: Simplifying expressions is crucial in engineering to design and optimize systems.
• Computer Science: Simplifying expressions is essential in computer science to optimize algorithms and data structures.

By mastering the skill of simplifying expressions, you can solve complex problems in various fields and make a significant impact in your career.