Multiply The Following Expression: \left(3x^2 - 2x\right)\left(2x^2 + 3x - 1\right ]A. 6 X 4 + 5 X 3 − 9 X 2 − 2 X 6x^4 + 5x^3 - 9x^2 - 2x 6 X 4 + 5 X 3 − 9 X 2 − 2 X B. 5 X 4 + 2 X 3 − 8 X 2 + 2 X 5x^4 + 2x^3 - 8x^2 + 2x 5 X 4 + 2 X 3 − 8 X 2 + 2 X C. 6 X 4 + 5 X 3 − 3 X 2 + 2 X 6x^4 + 5x^3 - 3x^2 + 2x 6 X 4 + 5 X 3 − 3 X 2 + 2 X D. 6 X 4 + 5 X 3 − 9 X 2 + 2 X 6x^4 + 5x^3 - 9x^2 + 2x 6 X 4 + 5 X 3 − 9 X 2 + 2 X

Introduction

Multiplying algebraic expressions is a fundamental concept in mathematics that involves combining like terms and applying the distributive property. In this article, we will focus on multiplying two quadratic expressions, which is a common scenario in algebra. We will use the given expression $\left(3x^2 - 2x\right)\left(2x^2 + 3x - 1\right)$ as an example and demonstrate how to multiply it using the distributive property.

The Distributive Property

The distributive property is a fundamental concept in algebra that 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 multiplying algebraic expressions.

Multiplying the Given Expression

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

Step 1: Multiply the First Term in the First Expression by Each Term in the Second Expression

We will start by multiplying the first term in the first expression, $3x^2$, by each term in the second expression.

$$3x^2(2x^2) = 6x^4$$

$$3x^2(3x) = 9x^3$$

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

Step 2: Multiply the Second Term in the First Expression by Each Term in the Second Expression

Next, we will multiply the second term in the first expression, $-2x$, by each term in the second expression.

$$-2x(2x^2) = -4x^3$$

$$-2x(3x) = -6x^2$$

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

Step 3: Combine Like Terms

Now, we will combine like terms to simplify the expression.

$$6x^4 + 9x^3 - 3x^2 - 4x^3 - 6x^2 + 2x$$

$$= 6x^4 + (9x^3 - 4x^3) + (-3x^2 - 6x^2) + 2x$$

$$= 6x^4 + 5x^3 - 9x^2 + 2x$$

Conclusion

In conclusion, multiplying the given expression $\left(3x^2 - 2x\right)\left(2x^2 + 3x - 1\right)$ using the distributive property results in the expression $6x^4 + 5x^3 - 9x^2 + 2x$. This is the correct answer among the options provided.

Answer

The correct answer is:

• D. $6x^4 + 5x^3 - 9x^2 + 2x$

Discussion

This problem requires the application of the distributive property to multiply two quadratic expressions. The distributive property is a fundamental concept in algebra that allows us to multiply expressions by combining like terms. In this problem, we multiplied each term in the first expression by each term in the second expression and then combined like terms to simplify the expression. This problem requires a strong understanding of the distributive property and the ability to apply it to complex expressions.

Tips and Tricks

When multiplying algebraic expressions, it is essential to apply the distributive property carefully. Make sure to multiply each term in the first expression by each term in the second expression and then combine like terms. This will ensure that you get the correct answer.

Common Mistakes

One common mistake when multiplying algebraic expressions is to forget to combine like terms. Make sure to combine like terms carefully to avoid this mistake.

Real-World Applications

Multiplying algebraic expressions has many real-world applications. For example, in physics, the motion of an object can be described using algebraic expressions. In economics, the cost of producing a product can be described using algebraic expressions. In engineering, the design of a system can be described using algebraic expressions.

Conclusion

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Introduction

Multiplying algebraic expressions is a fundamental concept in mathematics that involves combining like terms and applying the distributive property. In our previous article, we demonstrated how to multiply the given expression $\left(3x^2 - 2x\right)\left(2x^2 + 3x - 1\right)$ using the distributive property. In this article, we will provide a Q&A guide to help you understand the concept of multiplying algebraic expressions better.

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 $c$, 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 multiplying algebraic expressions.

Q: How do I apply the distributive property to multiply algebraic expressions?

A: To apply the distributive property, you need to multiply each term in the first expression by each term in the second expression and then combine like terms. This will ensure that you get the correct answer.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $2x^2$ and $3x^2$ are like terms because they have the same variable ($x$) and exponent ($2$).

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have $2x^2 + 3x^2$, you can combine them by adding the coefficients:

$$2x^2 + 3x^2 = 5x^2$$

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 combine like terms
• Not applying the distributive property correctly
• Not simplifying the expression after multiplying

Q: How do I simplify an expression after multiplying?

A: To simplify an expression after multiplying, you need to combine like terms and remove any unnecessary parentheses or brackets.

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

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

• Physics: The motion of an object can be described using algebraic expressions.
• Economics: The cost of producing a product can be described using algebraic expressions.
• Engineering: The design of a system can be described using algebraic expressions.

Q: How can I practice multiplying algebraic expressions?

A: You can practice multiplying algebraic expressions by:

• Working through examples and exercises in your textbook or online resources
• Creating your own examples and exercises
• Using online tools or software to practice multiplying algebraic expressions

Conclusion

In conclusion, multiplying algebraic expressions is a fundamental concept in mathematics that involves combining like terms and applying the distributive property. In this article, we provided a Q&A guide to help you understand the concept of multiplying algebraic expressions better. We also discussed some common mistakes to avoid and provided some real-world applications of multiplying algebraic expressions. By practicing multiplying algebraic expressions, you can improve your skills and become more confident in your ability to solve complex problems.

Tips and Tricks

• Make sure to apply the distributive property correctly when multiplying algebraic expressions.
• Combine like terms carefully to avoid mistakes.
• Simplify the expression after multiplying to make it easier to read and understand.

Common Mistakes

• Forgetting to combine like terms
• Not applying the distributive property correctly
• Not simplifying the expression after multiplying

Real-World Applications

• Physics: The motion of an object can be described using algebraic expressions.
• Economics: The cost of producing a product can be described using algebraic expressions.
• Engineering: The design of a system can be described using algebraic expressions.

Conclusion

In conclusion, multiplying algebraic expressions is a fundamental concept in mathematics that involves combining like terms and applying the distributive property. By practicing multiplying algebraic expressions, you can improve your skills and become more confident in your ability to solve complex problems.