Multiply: \[$(x-4)(x+5)\$\]

by ADMIN 28 views

Introduction

In algebra, multiplying expressions is a fundamental operation that helps us simplify complex equations and solve problems. In this article, we will focus on multiplying two binomial expressions, specifically {(x-4)(x+5)$}$. We will break down the process into manageable steps, making it easier to understand and apply.

What are Binomial Expressions?

A binomial expression is a polynomial with two terms. It can be written in the form {ax + b$}$, where {a$}$ and {b$}$ are constants, and {x$}$ is the variable. In our example, {(x-4)$}$ and {(x+5)$}$ are binomial expressions.

The FOIL Method

To multiply two binomial expressions, we can use the FOIL method. FOIL stands for "First, Outer, Inner, Last," which refers to the order in which we multiply the terms.

Step 1: Multiply the First Terms

The first term in the first expression is {x$}$, and the first term in the second expression is also {x$}$. We multiply these two terms together to get {x^2$}$.

Step 2: Multiply the Outer Terms

The outer terms are {x$}$ and {-4$}$. We multiply these two terms together to get −4x${-4x\$}.

Step 3: Multiply the Inner Terms

The inner terms are {-4$}$ and {x$}$. We multiply these two terms together to get −4x${-4x\$}.

Step 4: Multiply the Last Terms

The last terms are {-4$}$ and ${5\$}. We multiply these two terms together to get −20${-20\$}.

Combining the Terms

Now that we have multiplied all the terms, we can combine them to get the final result.

{(x-4)(x+5) = x^2 - 4x - 4x + 20$}$

We can simplify this expression by combining like terms.

{x^2 - 8x + 20$}$

Conclusion

Multiplying binomial expressions is a fundamental operation in algebra. By using the FOIL method, we can break down the process into manageable steps and simplify complex expressions. In this article, we multiplied the expressions {(x-4)(x+5)$}$ and obtained the final result {x^2 - 8x + 20$}$. We hope this article has helped you understand the process of multiplying binomial expressions and how to apply it to solve problems.

Example Problems

Problem 1

Multiply the expressions {(x+2)(x-3)$}$.

Solution

Using the FOIL method, we multiply the first terms {x$}$ and {x$}$ to get {x^2$}$. Then, we multiply the outer terms {x$}$ and {-3$}$ to get −3x${-3x\$}. Next, we multiply the inner terms ${2\$} and {x$}$ to get ${2x\$}. Finally, we multiply the last terms ${2\$} and {-3$}$ to get −6${-6\$}. Combining the terms, we get {x^2 - 3x + 2x - 6$}$. Simplifying the expression, we get {x^2 - x - 6$}$.

Problem 2

Multiply the expressions {(x-1)(x+4)$}$.

Solution

Using the FOIL method, we multiply the first terms {x$}$ and {x$}$ to get {x^2$}$. Then, we multiply the outer terms {x$}$ and ${4\$} to get ${4x\$}. Next, we multiply the inner terms {-1$}$ and {x$}$ to get −x${-x\$}. Finally, we multiply the last terms {-1$}$ and ${4\$} to get −4${-4\$}. Combining the terms, we get {x^2 + 4x - x - 4$}$. Simplifying the expression, we get {x^2 + 3x - 4$}$.

Tips and Tricks

  • When multiplying binomial expressions, always use the FOIL method to ensure that you multiply the terms in the correct order.
  • When combining like terms, make sure to combine the coefficients of the terms with the same variable.
  • When simplifying expressions, make sure to combine the terms in the correct order.

Conclusion

Introduction

In our previous article, we discussed the process of multiplying binomial expressions using the FOIL method. In this article, we will answer some frequently asked questions about multiplying algebraic expressions.

Q: What is the FOIL method?

A: The FOIL method is a technique used to multiply two binomial expressions. It stands for "First, Outer, Inner, Last," which refers to the order in which we multiply the terms.

Q: How do I use the FOIL method?

A: To use the FOIL method, follow these steps:

  1. Multiply the first terms of each expression.
  2. Multiply the outer terms of each expression.
  3. Multiply the inner terms of each expression.
  4. Multiply the last terms of each expression.
  5. Combine the terms to get the final result.

Q: What is the difference between multiplying binomial expressions and multiplying polynomial expressions?

A: Multiplying binomial expressions involves multiplying two expressions with two terms each, while multiplying polynomial expressions involves multiplying expressions with more than two terms.

Q: Can I use the FOIL method to multiply polynomial expressions?

A: No, the FOIL method is specifically designed for multiplying binomial expressions. To multiply polynomial expressions, you will need to use a different method, such as the distributive property.

Q: How do I simplify expressions after multiplying?

A: To simplify expressions after multiplying, combine like terms by adding or subtracting the coefficients of the terms with the same variable.

Q: What is the distributive property?

A: The distributive property is a technique used to multiply a single term by two or more terms. It states that a single term can be multiplied by each term in a group of terms separately.

Q: How do I use the distributive property?

A: To use the distributive property, follow these steps:

  1. Multiply the single term by each term in the group of terms.
  2. Combine the terms to get the final result.

Q: Can I use the FOIL method to multiply expressions with variables and constants?

A: Yes, the FOIL method can be used to multiply expressions with variables and constants. Simply treat the constants as if they were variables and follow the same steps as before.

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

A: Some common mistakes to avoid when multiplying algebraic expressions include:

  • Forgetting to multiply all the terms
  • Multiplying the terms in the wrong order
  • Not combining like terms
  • Not simplifying the expression after multiplying

Q: How can I practice multiplying algebraic expressions?

A: You can practice multiplying algebraic expressions by working through examples and exercises in a textbook or online resource. You can also try creating your own examples and solving them on your own.

Conclusion

Multiplying algebraic expressions is a fundamental operation in algebra. By using the FOIL method and the distributive property, you can simplify complex expressions and solve problems. In this article, we answered some frequently asked questions about multiplying algebraic expressions and provided tips and tricks for success. We hope this article has helped you understand the process of multiplying algebraic expressions and how to apply it to solve problems.

Additional Resources

Glossary

  • Binomial expression: An expression with two terms.
  • FOIL method: A technique used to multiply two binomial expressions.
  • Distributive property: A technique used to multiply a single term by two or more terms.
  • Like terms: Terms with the same variable and coefficient.