Simplify The Expression: { (n-2)(2n+1)$}$

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. It involves combining like terms, removing parentheses, and rearranging the expression to make it easier to work with. In this article, we will simplify the expression (nβˆ’2)(2n+1)(n-2)(2n+1) using the distributive property and other algebraic techniques.

The Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. In the expression (nβˆ’2)(2n+1)(n-2)(2n+1), we can use the distributive property to simplify it.

Expanding the Expression

To expand the expression, we multiply each term inside the parentheses with the term outside. This gives us:

(nβˆ’2)(2n+1)=n(2n+1)βˆ’2(2n+1)(n-2)(2n+1) = n(2n+1) - 2(2n+1)

Simplifying the Expression

Now, we can simplify the expression by combining like terms. We multiply each term inside the parentheses with the term outside and then combine like terms.

n(2n+1)βˆ’2(2n+1)=2n2+nβˆ’4nβˆ’2n(2n+1) - 2(2n+1) = 2n^2 + n - 4n - 2

Combining Like Terms

We can combine like terms by adding or subtracting the coefficients of the same variable. In this case, we have:

2n2+nβˆ’4nβˆ’2=2n2βˆ’3nβˆ’22n^2 + n - 4n - 2 = 2n^2 - 3n - 2

Alternative Method: FOIL Method

The FOIL method is another way to simplify the expression (nβˆ’2)(2n+1)(n-2)(2n+1). FOIL stands for First, Outer, Inner, Last, which refers to the order in which we multiply the terms.

Using the FOIL Method

To use the FOIL method, we multiply the first terms in each parentheses, then the outer terms, then the inner terms, and finally the last terms.

(nβˆ’2)(2n+1)=n(2n)+n(1)βˆ’2(2n)βˆ’2(1)(n-2)(2n+1) = n(2n) + n(1) - 2(2n) - 2(1)

Simplifying the Expression

Now, we can simplify the expression by combining like terms.

n(2n)+n(1)βˆ’2(2n)βˆ’2(1)=2n2+nβˆ’4nβˆ’2n(2n) + n(1) - 2(2n) - 2(1) = 2n^2 + n - 4n - 2

Combining Like Terms

We can combine like terms by adding or subtracting the coefficients of the same variable. In this case, we have:

2n2+nβˆ’4nβˆ’2=2n2βˆ’3nβˆ’22n^2 + n - 4n - 2 = 2n^2 - 3n - 2

Conclusion

In this article, we simplified the expression (nβˆ’2)(2n+1)(n-2)(2n+1) using the distributive property and the FOIL method. We showed that both methods produce the same result, which is 2n2βˆ’3nβˆ’22n^2 - 3n - 2. This expression can be used to solve equations and inequalities involving quadratic expressions.

Final Answer

The final answer is: 2n2βˆ’3nβˆ’2\boxed{2n^2 - 3n - 2}

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Introduction

In our previous article, we simplified the expression (nβˆ’2)(2n+1)(n-2)(2n+1) using the distributive property and the FOIL method. In this article, we will answer some frequently asked questions related to simplifying expressions like this one.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. It is often represented by the equation:

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

Q: How do I use the distributive property to simplify expressions?

A: To use the distributive property, you multiply each term inside the parentheses with the term outside. For example, in the expression (nβˆ’2)(2n+1)(n-2)(2n+1), you would multiply nn with 2n2n and 11, and then multiply βˆ’2-2 with 2n2n and 11.

Q: What is the FOIL method?

A: The FOIL method is a technique used to simplify expressions of the form (a+b)(c+d)(a+b)(c+d). It stands for First, Outer, Inner, Last, which refers to the order in which you multiply the terms.

Q: How do I use the FOIL method to simplify expressions?

A: To use the FOIL method, you multiply the first terms in each parentheses, then the outer terms, then the inner terms, and finally the last terms. For example, in the expression (nβˆ’2)(2n+1)(n-2)(2n+1), you would multiply nn with 2n2n, then nn with 11, then βˆ’2-2 with 2n2n, and finally βˆ’2-2 with 11.

Q: What is the difference between the distributive property and the FOIL method?

A: The distributive property is a general rule that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. The FOIL method is a specific technique used to simplify expressions of the form (a+b)(c+d)(a+b)(c+d).

Q: Can I use the FOIL method to simplify expressions that are not in the form (a+b)(c+d)(a+b)(c+d)?

A: No, the FOIL method is specifically designed to simplify expressions of the form (a+b)(c+d)(a+b)(c+d). If you have an expression that is not in this form, you should use the distributive property instead.

Q: How do I know which method to use?

A: If you have an expression that is in the form (a+b)(c+d)(a+b)(c+d), you can use the FOIL method. If you have an expression that is not in this form, you should use the distributive property.

Example Problems

Problem 1

Simplify the expression (x+3)(2xβˆ’1)(x+3)(2x-1) using the distributive property.

Solution

To simplify the expression, we multiply each term inside the parentheses with the term outside.

(x+3)(2xβˆ’1)=x(2x)+x(βˆ’1)+3(2x)+3(βˆ’1)(x+3)(2x-1) = x(2x) + x(-1) + 3(2x) + 3(-1)

Simplifying the Expression

Now, we can simplify the expression by combining like terms.

x(2x)+x(βˆ’1)+3(2x)+3(βˆ’1)=2x2βˆ’x+6xβˆ’3x(2x) + x(-1) + 3(2x) + 3(-1) = 2x^2 - x + 6x - 3

Combining Like Terms

We can combine like terms by adding or subtracting the coefficients of the same variable. In this case, we have:

2x2βˆ’x+6xβˆ’3=2x2+5xβˆ’32x^2 - x + 6x - 3 = 2x^2 + 5x - 3

Problem 2

Simplify the expression (xβˆ’2)(x+3)(x-2)(x+3) using the FOIL method.

Solution

To simplify the expression, we multiply the first terms in each parentheses, then the outer terms, then the inner terms, and finally the last terms.

(xβˆ’2)(x+3)=x(x)+x(3)βˆ’2(x)βˆ’2(3)(x-2)(x+3) = x(x) + x(3) - 2(x) - 2(3)

Simplifying the Expression

Now, we can simplify the expression by combining like terms.

x(x)+x(3)βˆ’2(x)βˆ’2(3)=x2+3xβˆ’2xβˆ’6x(x) + x(3) - 2(x) - 2(3) = x^2 + 3x - 2x - 6

Combining Like Terms

We can combine like terms by adding or subtracting the coefficients of the same variable. In this case, we have:

x2+3xβˆ’2xβˆ’6=x2+xβˆ’6x^2 + 3x - 2x - 6 = x^2 + x - 6

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions like (nβˆ’2)(2n+1)(n-2)(2n+1). We discussed the distributive property and the FOIL method, and provided example problems to illustrate how to use these techniques.

Final Answer

The final answer is: 2n2βˆ’3nβˆ’2\boxed{2n^2 - 3n - 2}