Which Expression Is Equal To $(x-3)\left(2 X^2-x+3\right)$?A. 2 X 3 + 5 X 2 − 9 2 X^3+5 X^2-9 2 X 3 + 5 X 2 − 9 B. 2 X 3 + 5 X 2 + 6 X − 9 2 X^3+5 X^2+6 X-9 2 X 3 + 5 X 2 + 6 X − 9 C. 2 X 3 − 7 X 2 − 9 2 X^3-7 X^2-9 2 X 3 − 7 X 2 − 9 D. 2 X 3 − 7 X 2 + 6 X − 9 2 X^3-7 X^2+6 X-9 2 X 3 − 7 X 2 + 6 X − 9

Mar 13, 2025 by ADMIN 305 views

**Expanding Algebraic Expressions: A Step-by-Step Guide**

===========================================================

Introduction

In algebra, expanding expressions is a crucial skill that helps us simplify complex equations and solve problems. In this article, we will focus on expanding the given expression $(x-3)\left(2 x^2-x+3\right)$ and determine which of the provided options is equal to it.

Understanding the Expression

The given expression is a product of two binomials: $(x-3)$ and $\left(2 x^2-x+3\right)$ . To expand this expression, we will use the distributive property, which states that for any real numbers $a$ , $b$ , and $c$ , $a(b+c) = ab + ac$ .

Expanding the Expression

To expand the given expression, we will multiply each term in the first binomial $(x-3)$ by each term in the second binomial $\left(2 x^2-x+3\right)$ . This will result in a sum of three terms.

Step 1: Multiply the First Term

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

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

Step 2: Multiply the Second Term

The second term in the first binomial is $-3$ . We will multiply this term by each term in the second binomial:

$-3 \cdot 2 x^2 = -6 x^2$ $-3 \cdot (-x) = 3 x$ $-3 \cdot 3 = -9$

Combining Like Terms

Now that we have multiplied each term in the first binomial by each term in the second binomial, we can combine like terms to simplify the expression:

$2 x^3 - x^2 + 3 x - 6 x^2 + 3 x - 9$

Combining like terms, we get:

$2 x^3 - 7 x^2 + 6 x - 9$

Conclusion

Based on our step-by-step expansion of the given expression, we can conclude that the correct answer is:

$2 x^3 - 7 x^2 + 6 x - 9$

This expression is equal to the original expression $(x-3)\left(2 x^2-x+3\right)$ .

Comparison with Options

Let's compare our result with the provided options:

A. $2 x^3+5 x^2-9$ B. $2 x^3+5 x^2+6 x-9$ C. $2 x^3-7 x^2-9$ D. $2 x^3-7 x^2+6 x-9$

Our result matches option D.

Final Answer

The final answer is:

D. $2 x^3-7 x^2+6 x-9$

This expression is equal to the original expression $(x-3)\left(2 x^2-x+3\right)$ .

Tips and Tricks

When expanding algebraic expressions, it's essential to follow the order of operations (PEMDAS):

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following this order of operations, you can ensure that you expand algebraic expressions correctly and accurately.

Practice Problems

To practice expanding algebraic expressions, try the following problems:

Expand the expression $(x+2)\left(x^2-3x+4\right)$ .
Expand the expression $(2x-1)\left(x^2+4x-3\right)$ .
Expand the expression $(x-1)\left(2x^2-5x+2\right)$ .

Remember to follow the order of operations and combine like terms to simplify the expression.

Conclusion

Expanding algebraic expressions is a crucial skill in algebra that helps us simplify complex equations and solve problems. By following the distributive property and combining like terms, we can expand expressions accurately and efficiently. In this article, we expanded the expression $(x-3)\left(2 x^2-x+3\right)$ and determined that the correct answer is $2 x^3-7 x^2+6 x-9$ . We also provided tips and tricks for expanding algebraic expressions and practice problems to help you practice your skills.

===========================================================

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra 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 two or more terms inside parentheses.

Q: How do I expand an algebraic expression?

A: To expand an algebraic expression, follow these steps:

Multiply each term in the first binomial by each term in the second binomial.
Combine like terms to simplify the expression.

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, $2x^2$ and $5x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms. For example, $2x^2 + 5x^2 = 7x^2$ .

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, follow these steps:

Expand the expression using the distributive property.
Combine like terms to simplify the expression.

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

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

Forgetting to multiply each term in the first binomial by each term in the second binomial.
Not combining like terms.
Not following the order of operations.

Q: How can I practice expanding algebraic expressions?

A: You can practice expanding algebraic expressions by working through practice problems, such as the ones listed below:

Expand the expression $(x+2)\left(x^2-3x+4\right)$ .
Expand the expression $(2x-1)\left(x^2+4x-3\right)$ .
Expand the expression $(x-1)\left(2x^2-5x+2\right)$ .

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

A: Expanding algebraic expressions has many real-world applications, including:

Solving systems of equations in physics and engineering.
Modeling population growth and decay in biology.
Analyzing data in statistics.

Q: How can I use technology to help me expand algebraic expressions?

A: You can use technology, such as graphing calculators or computer algebra systems, to help you expand algebraic expressions. These tools can perform calculations and simplify expressions for you.

Q: What are some tips for expanding algebraic expressions?

A: Some tips for expanding algebraic expressions include:

Use the distributive property to expand expressions.
Combine like terms to simplify expressions.
Follow the order of operations.
Practice, practice, practice!

Conclusion

Expanding algebraic expressions is a crucial skill in algebra that helps us simplify complex equations and solve problems. By following the distributive property and combining like terms, we can expand expressions accurately and efficiently. In this article, we answered frequently asked questions about expanding algebraic expressions and provided tips and tricks for expanding expressions. We also provided practice problems to help you practice your skills.