Which Expression Is Equivalent To $(2x+1)(3x^2-4x+5)$?A. $5x^3+2x^2+4x+6$B. $5x^3+2x^2+10x+6$C. $6x^3-5x^2+6x+5$D. $6x^3-5x^2+7x+5$

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Introduction

Multiplying algebraic expressions is a fundamental concept in algebra that can seem daunting at first, but with practice and a clear understanding of the steps involved, it becomes a manageable task. In this article, we will explore the process of multiplying two algebraic expressions, using the given expression (2x+1)(3x2βˆ’4x+5)(2x+1)(3x^2-4x+5) as an example. We will also examine the different options provided and determine which one is equivalent to the given expression.

The FOIL Method

The FOIL method is a popular technique used to multiply two binomials. FOIL stands for First, Outer, Inner, Last, which refers to the order in which the terms are multiplied. To apply the FOIL method, we multiply the first terms of each binomial, then the outer terms, followed by the inner terms, and finally the last terms.

Step 1: Multiply the First Terms

The first terms of the given binomials are 2x2x and 3x23x^2. Multiplying these terms gives us 6x36x^3.

Step 2: Multiply the Outer Terms

The outer terms are 2x2x and βˆ’4x-4x. Multiplying these terms gives us βˆ’8x2-8x^2.

Step 3: Multiply the Inner Terms

The inner terms are 11 and βˆ’4x-4x. Multiplying these terms gives us βˆ’4x-4x.

Step 4: Multiply the Last Terms

The last terms are 11 and 55. Multiplying these terms gives us 55.

Combining Like Terms

Now that we have multiplied all the terms, we need to combine like terms. This involves adding or subtracting terms that have the same variable and exponent.

Step 1: Combine the x3x^3 Terms

We have two terms with an exponent of 33: 6x36x^3 and 0x30x^3. Since there is no x3x^3 term in the second binomial, we can ignore it. The x3x^3 term remains as 6x36x^3.

Step 2: Combine the x2x^2 Terms

We have two terms with an exponent of 22: βˆ’8x2-8x^2 and 0x20x^2. Since there is no x2x^2 term in the second binomial, we can ignore it. The x2x^2 term remains as βˆ’8x2-8x^2.

Step 3: Combine the xx Terms

We have two terms with an exponent of 11: βˆ’4x-4x and 2x2x. Adding these terms gives us βˆ’2x-2x.

Step 4: Combine the Constant Terms

We have two constant terms: 55 and 00. Since there is no constant term in the second binomial, we can ignore it. The constant term remains as 55.

The Final Expression

Combining all the terms, we get the final expression: 6x3βˆ’8x2βˆ’2x+56x^3-8x^2-2x+5.

Comparing with the Options

Now that we have the final expression, we can compare it with the options provided.

Option A: 5x3+2x2+4x+65x^3+2x^2+4x+6

This option is not equivalent to the given expression. The x3x^3 term is incorrect, and the x2x^2 term is also incorrect.

Option B: 5x3+2x2+10x+65x^3+2x^2+10x+6

This option is not equivalent to the given expression. The x3x^3 term is incorrect, and the x2x^2 term is also incorrect.

Option C: 6x3βˆ’5x2+6x+56x^3-5x^2+6x+5

This option is not equivalent to the given expression. The x2x^2 term is incorrect, and the xx term is also incorrect.

Option D: 6x3βˆ’5x2+7x+56x^3-5x^2+7x+5

This option is not equivalent to the given expression. The xx term is incorrect.

Conclusion

In conclusion, the correct answer is not among the options provided. The final expression is 6x3βˆ’8x2βˆ’2x+56x^3-8x^2-2x+5, which is not equivalent to any of the options.

However, if we re-examine the options, we can see that option C is close to the correct answer. The only difference is the xx term, which is 6x6x instead of βˆ’2x-2x. This suggests that the correct answer may be a variation of option C.

The Correct Answer

After re-examining the options, we can see that the correct answer is actually a variation of option C. The correct answer is 6x3βˆ’5x2+6x+56x^3-5x^2+6x+5, which is equivalent to the given expression.

However, this answer is not among the options provided. The closest option is option C, which is 6x3βˆ’5x2+6x+56x^3-5x^2+6x+5. This option is equivalent to the given expression, but it is not the correct answer.

The Final Answer

The final answer is 6x3βˆ’5x2+6x+56x^3-5x^2+6x+5.

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Introduction

Multiplying algebraic expressions is a fundamental concept in algebra that can seem daunting at first, but with practice and a clear understanding of the steps involved, it becomes a manageable task. In this article, we will explore the process of multiplying two algebraic expressions, using the given expression (2x+1)(3x2βˆ’4x+5)(2x+1)(3x^2-4x+5) as an example. We will also examine the different options provided and determine which one is equivalent to the given expression.

Q&A: Multiplying Algebraic Expressions

Q: What is the FOIL method?

A: The FOIL method is a popular technique used to multiply two binomials. FOIL stands for First, Outer, Inner, Last, which refers to the order in which the terms are multiplied.

Q: How do I apply the FOIL method?

A: To apply the FOIL method, you multiply the first terms of each binomial, then the outer terms, followed by the inner terms, and finally the last terms.

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

A: The FOIL method is a specific technique used to multiply two binomials, while the distributive property is a general rule that can be applied to any expression. The distributive property states that a single term can be multiplied by each term in a binomial.

Q: How do I combine like terms?

A: To combine like terms, you add or subtract terms that have the same variable and exponent.

Q: What is the final expression for the given expression (2x+1)(3x2βˆ’4x+5)(2x+1)(3x^2-4x+5)?

A: The final expression for the given expression is 6x3βˆ’8x2βˆ’2x+56x^3-8x^2-2x+5.

Q: Which option is equivalent to the given expression?

A: The correct answer is actually a variation of option C. The correct answer is 6x3βˆ’5x2+6x+56x^3-5x^2+6x+5, which is equivalent to the given expression.

Common Mistakes to Avoid

Mistake 1: Not applying the FOIL method correctly

A: Make sure to multiply the first terms of each binomial, then the outer terms, followed by the inner terms, and finally the last terms.

Mistake 2: Not combining like terms correctly

A: Make sure to add or subtract terms that have the same variable and exponent.

Mistake 3: Not checking the options carefully

A: Make sure to carefully examine each option and determine which one is equivalent to the given expression.

Conclusion

In conclusion, multiplying algebraic expressions can seem daunting at first, but with practice and a clear understanding of the steps involved, it becomes a manageable task. By applying the FOIL method and combining like terms correctly, you can determine which option is equivalent to the given expression.

Final Tips

Tip 1: Practice, practice, practice

A: The more you practice multiplying algebraic expressions, the more comfortable you will become with the process.

Tip 2: Use the distributive property

A: The distributive property can be a helpful tool when multiplying algebraic expressions.

Tip 3: Check your work carefully

A: Make sure to carefully examine each option and determine which one is equivalent to the given expression.

Additional Resources

Online Resources

A: There are many online resources available that can help you learn how to multiply algebraic expressions, including video tutorials and practice problems.

Textbooks

A: There are many textbooks available that can help you learn how to multiply algebraic expressions, including algebra textbooks and online resources.

Practice Problems

A: There are many practice problems available that can help you learn how to multiply algebraic expressions, including online resources and worksheets.