Forgetting to multiply each term in the first expression by each term in the second expression

Oct 13, 2025 by wbrum.org 107 views

Introduction

In algebra, expanding an expression to a polynomial in standard form is a crucial step in solving equations and manipulating mathematical expressions. In this article, we will focus on expanding the given expression $\left(2x^2 + x - 2\right)\left(3x^2 + 4x + 1\right)$ to a polynomial in standard form. We will use the distributive property and the FOIL method to expand the expression.

The Distributive Property

The distributive property states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

a(b + c) = ab + ac

This property can be used to expand expressions by multiplying each term in the first expression by each term in the second expression.

The FOIL Method

The FOIL method is a technique used to expand expressions of the form $(a + b)(c + d)$ . The FOIL method stands for "First, Outer, Inner, Last," which refers to the order in which the terms are multiplied.

First: Multiply the first term in the first expression by the first term in the second expression.
Outer: Multiply the first term in the first expression by the second term in the second expression.
Inner: Multiply the second term in the first expression by the first term in the second expression.
Last: Multiply the second term in the first expression by the second term in the second expression.

Expanding the Expression

Using the distributive property and the FOIL method, we can expand the given expression as follows:

\left(2x^2 + x - 2\right)\left(3x^2 + 4x + 1\right)

First, we multiply the first term in the first expression by each term in the second expression:

2x^2(3x^2) + 2x^2(4x) + 2x^2(1)

Next, we multiply the second term in the first expression by each term in the second expression:

x(3x^2) + x(4x) + x(1)

Finally, we multiply the third term in the first expression by each term in the second expression:

-2(3x^2) - 2(4x) - 2(1)

Now, we can combine like terms and simplify the expression:

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

Combining like terms, we get:

6x^4 + (8x^3 + 3x^3) + (2x^2 + 4x^2 - 6x^2) + (x - 8x) - 2

Simplifying further, we get:

6x^4 + 11x^3 - 0x^2 - 7x - 2

Therefore, the expanded expression in standard form is:

6x^4 + 11x^3 - 0x^2 - 7x - 2

Conclusion

In this article, we expanded the given expression $\left(2x^2 + x - 2\right)\left(3x^2 + 4x + 1\right)$ to a polynomial in standard form using the distributive property and the FOIL method. We multiplied each term in the first expression by each term in the second expression and combined like terms to simplify the expression. The expanded expression in standard form is $6x^4 + 11x^3 - 0x^2 - 7x - 2$ . This expression can be used to solve equations and manipulate mathematical expressions.

Applications

The expanded expression can be used in various applications, such as:

Solving equations: The expanded expression can be used to solve equations by setting the expression equal to zero and solving for the variable.
Graphing functions: The expanded expression can be used to graph functions by plotting the expression on a coordinate plane.
Optimization: The expanded expression can be used to optimize functions by finding the maximum or minimum value of the expression.

Final Thoughts

Introduction

In our previous article, we expanded the given expression $\left(2x^2 + x - 2\right)\left(3x^2 + 4x + 1\right)$ to a polynomial in standard form using the distributive property and the FOIL method. In this article, we will answer some frequently asked questions about expanding expressions to polynomials in standard form.

Q: What is the distributive property?

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

a(b + c) = ab + ac

This property can be used to expand expressions by multiplying each term in the first expression by each term in the second expression.

Q: What is the FOIL method?

A: The FOIL method is a technique used to expand expressions of the form $(a + b)(c + d)$ . The FOIL method stands for "First, Outer, Inner, Last," which refers to the order in which the terms are multiplied.

First: Multiply the first term in the first expression by the first term in the second expression.
Outer: Multiply the first term in the first expression by the second term in the second expression.
Inner: Multiply the second term in the first expression by the first term in the second expression.
Last: Multiply the second term in the first expression by the second term in the second expression.

Q: How do I expand an expression using the distributive property and the FOIL method?

A: To expand an expression using the distributive property and the FOIL method, follow these steps:

Multiply each term in the first expression by each term in the second expression.
Combine like terms to simplify the expression.
Use the FOIL method to expand expressions of the form $(a + b)(c + d)$ .

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

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

Forgetting to multiply each term in the first expression by each term in the second expression.
Not combining like terms to simplify the expression.
Using the FOIL method incorrectly.

Q: How do I check my work when expanding an expression?

A: To check your work when expanding an expression, follow these steps:

Multiply each term in the first expression by each term in the second expression.
Combine like terms to simplify the expression.
Use the FOIL method to expand expressions of the form $(a + b)(c + d)$ .
Check your work by plugging in values for the variable and simplifying the expression.

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

A: Some real-world applications of expanding expressions include:

Solving equations: Expanding expressions can be used to solve equations by setting the expression equal to zero and solving for the variable.
Graphing functions: Expanding expressions can be used to graph functions by plotting the expression on a coordinate plane.
Optimization: Expanding expressions can be used to optimize functions by finding the maximum or minimum value of the expression.

Conclusion

In conclusion, expanding an expression to a polynomial in standard form is a crucial step in solving equations and manipulating mathematical expressions. The distributive property and the FOIL method are useful techniques for expanding expressions. By following the steps outlined in this article, you can expand expressions and check your work to ensure accuracy.

Final Thoughts

Expanding expressions is a fundamental concept in algebra that has numerous real-world applications. By mastering the distributive property and the FOIL method, you can solve equations, graph functions, and optimize functions with ease. Remember to check your work and avoid common mistakes to ensure accuracy.