Evaluate The Expression: $ (2x^2 + 3x - 4)(x - 2) = $

Introduction

In algebra, evaluating expressions is a crucial skill that helps us simplify complex mathematical statements. When we encounter an expression like $(2x^2 + 3x - 4)(x - 2)$, our goal is to simplify it by multiplying the two binomials. This process involves using the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. In this article, we will walk through the step-by-step process of evaluating the given expression.

Understanding the Expression

Before we dive into the evaluation process, let's break down the given expression and understand its components. The expression is a product of two binomials:

$(2x^2 + 3x - 4)(x - 2)$

The first binomial is a quadratic expression with a leading coefficient of 2, while the second binomial is a linear expression with a coefficient of 1. Our task is to multiply these two binomials using the distributive property.

Step 1: Multiply the First Binomial by the Second Binomial

To multiply the two binomials, we will use the distributive property. We will multiply each term in the first binomial by each term in the second binomial. This process involves multiplying the leading term of the first binomial by the entire second binomial, and then multiplying the remaining terms of the first binomial by the entire second binomial.

Let's start by multiplying the first term of the first binomial, $2x^2$, by the entire second binomial, $(x - 2)$.

$2x^2(x - 2) = 2x^3 - 4x^2$

Next, we will multiply the second term of the first binomial, $3x$, by the entire second binomial, $(x - 2)$.

$3x(x - 2) = 3x^2 - 6x$

Finally, we will multiply the third term of the first binomial, $-4$, by the entire second binomial, $(x - 2)$.

$-4(x - 2) = -4x + 8$

Step 2: Combine Like Terms

Now that we have multiplied the two binomials, we need to combine like terms. This involves adding or subtracting terms with the same variable and exponent.

Let's combine the terms we obtained in the previous step:

$2x^3 - 4x^2 + 3x^2 - 6x - 4x + 8$

We can combine the like terms $-4x^2$ and $3x^2$ to get $-x^2$. We can also combine the like terms $-6x$ and $-4x$ to get $-10x$.

The final expression is:

$2x^3 - x^2 - 10x + 8$

Conclusion

In this article, we evaluated the expression $(2x^2 + 3x - 4)(x - 2)$ by multiplying the two binomials using the distributive property. We broke down the expression into smaller components, multiplied each term in the first binomial by each term in the second binomial, and then combined like terms to simplify the expression. The final expression is $2x^3 - x^2 - 10x + 8$.

Tips and Tricks

• When multiplying binomials, use the distributive property to multiply each term in the first binomial by each term in the second binomial.
• Combine like terms to simplify the expression.
• Use the FOIL method to multiply two binomials: First, Outer, Inner, Last.

Real-World Applications

Evaluating expressions like $(2x^2 + 3x - 4)(x - 2)$ has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, we use algebraic expressions to model the motion of objects. In engineering, we use algebraic expressions to design and optimize systems. In economics, we use algebraic expressions to model economic systems and make predictions about future trends.

Common Mistakes

• Failing to use the distributive property when multiplying binomials.
• Failing to combine like terms.
• Making errors when multiplying terms.

Practice Problems

• Evaluate the expression $(x^2 + 2x - 3)(x + 1)$.
• Evaluate the expression $(2x^2 - 3x + 1)(x - 2)$.
• Evaluate the expression $(x^2 + 4x - 5)(x + 3)$.

Conclusion

Evaluating expressions like $(2x^2 + 3x - 4)(x - 2)$ is an essential skill in algebra. By using the distributive property and combining like terms, we can simplify complex expressions and make predictions about real-world phenomena. In this article, we walked through the step-by-step process of evaluating the given expression and provided tips and tricks for simplifying expressions. We also discussed real-world applications and common mistakes to avoid.

Introduction

In our previous article, we evaluated the expression $(2x^2 + 3x - 4)(x - 2)$ by multiplying the two binomials using the distributive property. We broke down the expression into smaller components, multiplied each term in the first binomial by each term in the second binomial, and then combined like terms to simplify the expression. In this article, we will answer some frequently asked questions about evaluating expressions like $(2x^2 + 3x - 4)(x - 2)$.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This means that we can multiply a single term by a sum of terms by multiplying the single term by each term in the sum.

Q: How do I multiply two binomials using the distributive property?

A: To multiply two binomials using the distributive property, we multiply each term in the first binomial by each term in the second binomial. This involves multiplying the leading term of the first binomial by the entire second binomial, and then multiplying the remaining terms of the first binomial by the entire second binomial.

Q: What is the FOIL method?

A: The FOIL method is a shortcut for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, and it refers to the order in which we multiply the terms in the two binomials. The FOIL method is a useful tool for multiplying binomials, but it is not a substitute for the distributive property.

Q: How do I combine like terms?

A: To combine like terms, we add or subtract terms with the same variable and exponent. For example, if we have the expression $2x^2 + 3x^2$, we can combine the like terms to get $5x^2$.

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

A: Some common mistakes to avoid when evaluating expressions include failing to use the distributive property, failing to combine like terms, and making errors when multiplying terms.

Q: How do I evaluate an expression with multiple binomials?

A: To evaluate an expression with multiple binomials, we can use the distributive property to multiply each binomial by each other binomial. This involves multiplying each term in the first binomial by each term in the second binomial, and then multiplying the result by each term in the third binomial, and so on.

Q: Can I use the FOIL method to evaluate expressions with multiple binomials?

A: No, the FOIL method is only suitable for multiplying two binomials. If we have an expression with multiple binomials, we need to use the distributive property to multiply each binomial by each other binomial.

Tips and Tricks

• Use the distributive property to multiply binomials.
• Combine like terms to simplify the expression.
• Use the FOIL method to multiply two binomials.
• Be careful when multiplying terms to avoid errors.

Real-World Applications

Evaluating expressions like $(2x^2 + 3x - 4)(x - 2)$ has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, we use algebraic expressions to model the motion of objects. In engineering, we use algebraic expressions to design and optimize systems. In economics, we use algebraic expressions to model economic systems and make predictions about future trends.

Practice Problems

• Evaluate the expression $(x^2 + 2x - 3)(x + 1)$.
• Evaluate the expression $(2x^2 - 3x + 1)(x - 2)$.
• Evaluate the expression $(x^2 + 4x - 5)(x + 3)$.

Conclusion

Evaluating expressions like $(2x^2 + 3x - 4)(x - 2)$ is an essential skill in algebra. By using the distributive property and combining like terms, we can simplify complex expressions and make predictions about real-world phenomena. In this article, we answered some frequently asked questions about evaluating expressions and provided tips and tricks for simplifying expressions. We also discussed real-world applications and common mistakes to avoid.