Selecting the Correct Product

Oct 13, 2025 by leolavanderia.com.br 110 views

Understanding the Problem

When multiplying polynomials, it's essential to follow the correct order of operations to obtain the correct product. In this article, we will focus on multiplying two binomials, $\left(x^2+3\right)$ and $\left(x^3-5\right)$ , and select the correct product from the given options.

The FOIL Method

To multiply two binomials, we can use the FOIL method, which stands for First, Outer, Inner, Last. This method helps us to multiply the terms in the correct order.

Step 1: Multiply the First Terms

The first term in the first binomial is $x^2$ , and the first term in the second binomial is $x^3$ . Multiplying these two terms, we get:

$x^2 \cdot x^3 = x^5$

Step 2: Multiply the Outer Terms

The outer terms are $x^2$ and $-5$ . Multiplying these two terms, we get:

$x^2 \cdot -5 = -5x^2$

Step 3: Multiply the Inner Terms

The inner terms are $3$ and $x^3$ . Multiplying these two terms, we get:

$3 \cdot x^3 = 3x^3$

Step 4: Multiply the Last Terms

The last terms are $3$ and $-5$ . Multiplying these two terms, we get:

$3 \cdot -5 = -15$

Combining the Terms

Now that we have multiplied all the terms, we can combine them to get the final product:

$x^5 - 5x^2 + 3x^3 - 15$

Selecting the Correct Product

Comparing the final product with the given options, we can see that the correct product is:

$x^5 - 5x^2 + 3x^3 - 15$

However, this option is not available in the given choices. Let's re-examine the options and see if we can find a match.

Option A: $x^5 - 15$

This option is missing the terms $-5x^2$ and $3x^3$ , which are present in the final product.

Option B: $x^5 + 3x^3 - 5x^2 - 15$

This option is missing the negative sign in front of the term $-5x^2$ , and the term $-15$ is present in the final product.

Option C: $x^6 - 15$

This option is missing the terms $-5x^2$ and $3x^3$ , and the term $x^6$ is not present in the final product.

Option D: $x^6 + 3x^3 - 5x^2 - 15$

This option is missing the term $x^5$ , and the term $x^6$ is not present in the final product.

Conclusion

After re-examining the options, we can see that none of the options match the final product. However, if we add the missing term $x^5$ to option B, we get:

$x^5 + 3x^3 - 5x^2 - 15$

This option matches the final product, and therefore, it is the correct answer.

Final Answer

The final answer is: $\boxed{B}$

Understanding the Problem

Multiplying polynomials can be a challenging task, especially when dealing with binomials. In this article, we will provide a Q&A guide to help you understand the process of multiplying polynomials and select the correct product.

Q: What is the FOIL method?

A: The FOIL method is a technique used to multiply two binomials. It stands for First, Outer, Inner, Last, and it helps us to multiply the terms in the correct order.

Q: How do I apply the FOIL method?

A: To apply the FOIL method, you need to multiply the first terms, then the outer terms, then the inner terms, and finally the last terms. This will give you the correct product.

Q: What are the steps involved in multiplying polynomials?

A: The steps involved in multiplying polynomials are:

Multiply the first terms
Multiply the outer terms
Multiply the inner terms
Multiply the last terms
Combine the terms to get the final product

Q: What is the difference between multiplying polynomials and multiplying numbers?

A: When multiplying polynomials, you need to multiply the terms in the correct order, using the FOIL method. When multiplying numbers, you can simply multiply the numbers together.

Q: How do I select the correct product from the given options?

A: To select the correct product, you need to compare the final product with the given options. Make sure to check for any missing terms or incorrect signs.

Q: What if I get a different product than the one listed in the options?

A: If you get a different product than the one listed in the options, it's possible that the options are incorrect or that you made a mistake in your calculations. Double-check your work and make sure to follow the correct order of operations.

Q: Can I use the FOIL method to multiply polynomials with more than two terms?

A: No, the FOIL method is only used to multiply two binomials. If you need to multiply polynomials with more than two terms, you will need to use a different method, such as the distributive property.

Q: How do I multiply polynomials with negative exponents?

A: When multiplying polynomials with negative exponents, you need to follow the same rules as when multiplying numbers with negative exponents. For example, $x^{-2} \cdot x^3 = x^{3-2} = x^1 = x$ .

Q: Can I use a calculator to multiply polynomials?

A: Yes, you can use a calculator to multiply polynomials. However, make sure to check your work and understand the process of multiplying polynomials, as calculators can sometimes make mistakes.

Conclusion

Multiplying polynomials can be a challenging task, but with the right techniques and understanding, you can select the correct product from the given options. Remember to follow the FOIL method, check for any missing terms or incorrect signs, and double-check your work.

Final Tips

Make sure to follow the correct order of operations when multiplying polynomials.
Use the FOIL method to multiply two binomials.
Check for any missing terms or incorrect signs when selecting the correct product.
Double-check your work and make sure to understand the process of multiplying polynomials.

Common Mistakes

Forgetting to multiply the first terms
Forgetting to multiply the outer terms
Forgetting to multiply the inner terms
Forgetting to multiply the last terms
Not checking for any missing terms or incorrect signs

Practice Problems

Multiply the following polynomials: $\left(x^2+3\right)\left(x^3-5\right)$
Multiply the following polynomials: $\left(x^2-2\right)\left(x^3+4\right)$
Multiply the following polynomials: $\left(x^2+1\right)\left(x^3-2\right)$

Answer Key

$\left(x^2+3\right)\left(x^3-5\right) = x^5 - 5x^2 + 3x^3 - 15$
$\left(x^2-2\right)\left(x^3+4\right) = x^5 + 2x^3 - 8x^2 - 8$
$\left(x^2+1\right)\left(x^3-2\right) = x^5 - 2x^2 + x^3 - 2$