Expand The Expression:a. { (x+1)\left(2x^2-3\right)$}$

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Introduction

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In algebra, expanding an expression involves multiplying the terms of two or more polynomials. This process is essential in solving equations, simplifying expressions, and factoring polynomials. In this article, we will focus on expanding the given expression: $(x+1)(2x^2-3)$. We will use the distributive property to multiply the terms and simplify the resulting expression.

The Distributive Property

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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 more than two terms, and it is the key to expanding expressions like the one given.

Expanding the Expression

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To expand the expression $(x+1)(2x^2-3)$, we will use the distributive property. We will multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.

Step 1: Multiply the First Term

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The first term in the first polynomial is $x$. We will multiply this term by each term in the second polynomial:

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

Step 2: Multiply the Second Term

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The second term in the first polynomial is $1$. We will multiply this term by each term in the second polynomial:

$1(2x^2-3) = 2x^2 - 3$

Step 3: Combine Like Terms

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Now that we have multiplied each term in the first polynomial by each term in the second polynomial, we can combine like terms. We will add or subtract the terms with the same variable and exponent:

$2x^3 - 3x + 2x^2 - 3$

Simplifying the Expression

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The expression we obtained in the previous step is not in its simplest form. We can simplify it by combining like terms:

$2x^3 + 2x^2 - 3x - 3$

Final Answer

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The expanded expression is:

$2x^3 + 2x^2 - 3x - 3$

Conclusion

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In this article, we expanded the given expression $(x+1)(2x^2-3)$ using the distributive property. We multiplied each term in the first polynomial by each term in the second polynomial and then combined like terms. The resulting expression is in its simplest form, and it can be used to solve equations, simplify expressions, and factor polynomials.

Example Use Cases

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The expanded expression can be used in a variety of mathematical contexts, including:

• Solving quadratic equations
• Simplifying rational expressions
• Factoring polynomials
• Finding the roots of a polynomial

Tips and Tricks

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When expanding expressions, it is essential to use the distributive property and combine like terms. This will help you to simplify the expression and make it easier to work with.

• Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.
• Combine like terms by adding or subtracting the terms with the same variable and exponent.
• Simplify the expression by combining like terms and eliminating any unnecessary terms.

Common Mistakes

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When expanding expressions, it is easy to make mistakes. Here are some common mistakes to avoid:

• Failing to use the distributive property
• Not combining like terms
• Introducing unnecessary terms
• Not simplifying the expression

Final Thoughts

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Expanding expressions is an essential skill in algebra, and it requires practice and patience. By using the distributive property and combining like terms, you can simplify expressions and make them easier to work with. Remember to use the distributive property, combine like terms, and simplify the expression to get the correct answer.

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Introduction

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In our previous article, we expanded the given expression $(x+1)(2x^2-3)$ using the distributive property. We multiplied each term in the first polynomial by each term in the second polynomial and then combined like terms. In this article, we will answer some frequently asked questions about expanding expressions.

Q&A

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Q: What is the distributive property?

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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 more than two terms, and it is the key to expanding expressions like the one given.

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

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A: To expand an expression using the distributive property, you need to multiply each term in the first polynomial by each term in the second polynomial and then combine like terms. Here's a step-by-step guide:

1. Multiply each term in the first polynomial by each term in the second polynomial.
2. Combine like terms by adding or subtracting the terms with the same variable and exponent.
3. Simplify the expression by eliminating any unnecessary terms.

Q: What are like terms?

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A: Like terms are terms that have the same variable and exponent. For example, $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ and the exponent $2$. When combining like terms, you add or subtract the coefficients of the like terms.

Q: How do I simplify an expression?

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A: To simplify an expression, you need to combine like terms and eliminate any unnecessary terms. Here's a step-by-step guide:

1. Combine like terms by adding or subtracting the terms with the same variable and exponent.
2. Eliminate any unnecessary terms by simplifying the expression.

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

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A: Here are some common mistakes to avoid when expanding expressions:

• Failing to use the distributive property
• Not combining like terms
• Introducing unnecessary terms
• Not simplifying the expression

Q: How do I check my work when expanding expressions?

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A: To check your work when expanding expressions, you need to follow these steps:

1. Multiply each term in the first polynomial by each term in the second polynomial.
2. Combine like terms by adding or subtracting the terms with the same variable and exponent.
3. Simplify the expression by eliminating any unnecessary terms.
4. Check your work by plugging in a value for the variable and simplifying the expression.

Example Problems

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Here are some example problems to help you practice expanding expressions:

Problem 1

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Expand the expression $(x+2)(x^2-3)$ using the distributive property.

Solution

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To expand the expression, we need to multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.

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

$= x^3 - 3x + 2x^2 - 6$

$= 2x^2 + x^3 - 3x - 6$

Problem 2

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Expand the expression $(x-1)(x^2+2)$ using the distributive property.

Solution

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To expand the expression, we need to multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.

$(x-1)(x^2+2) = x(x^2+2) - 1(x^2+2)$

$= x^3 + 2x - x^2 - 2$

$= x^3 - x^2 + 2x - 2$

Conclusion

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Expanding expressions is an essential skill in algebra, and it requires practice and patience. By using the distributive property and combining like terms, you can simplify expressions and make them easier to work with. Remember to use the distributive property, combine like terms, and simplify the expression to get the correct answer.