What Is The Product Of $(3a+2)\left(4a^2-2a+9\right)$?$A. $12a^3 - 2a + 18$B. $12a^3 + 6a + 9$C. $12a^3 - 6a^2 + 23a + 18$D. $12a^3 + 2a^2 + 23a + 18$

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Introduction

In algebra, the product of two or more polynomials is a fundamental concept that is used to simplify complex expressions. When multiplying polynomials, we need to apply the distributive property, which states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac. In this article, we will use the distributive property to find the product of the given polynomials (3a+2)(4a2−2a+9)(3a+2)\left(4a^2-2a+9\right).

Step 1: Apply the Distributive Property

To find the product of the given polynomials, we need to apply the distributive property. This means that we need to multiply each term in the first polynomial by each term in the second polynomial.

(3a+2)\left(4a^2-2a+9\right) = (3a)(4a^2) + (3a)(-2a) + (3a)(9) + (2)(4a^2) + (2)(-2a) + (2)(9)

Step 2: Simplify the Expression

Now that we have applied the distributive property, we can simplify the expression by combining like terms.

(3a)(4a^2) + (3a)(-2a) + (3a)(9) + (2)(4a^2) + (2)(-2a) + (2)(9) = 12a^3 - 6a^2 + 27a + 8a^2 - 4a + 18

Step 3: Combine Like Terms

We can further simplify the expression by combining like terms.

12a^3 - 6a^2 + 27a + 8a^2 - 4a + 18 = 12a^3 + 2a^2 + 23a + 18

Conclusion

In conclusion, the product of (3a+2)(4a2−2a+9)(3a+2)\left(4a^2-2a+9\right) is 12a3+2a2+23a+1812a^3 + 2a^2 + 23a + 18. This is the correct answer.

Answer

The correct answer is D. 12a3+2a2+23a+1812a^3 + 2a^2 + 23a + 18.

Discussion

The product of two or more polynomials is a fundamental concept in algebra that is used to simplify complex expressions. When multiplying polynomials, we need to apply the distributive property, which states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac. In this article, we used the distributive property to find the product of the given polynomials (3a+2)(4a2−2a+9)(3a+2)\left(4a^2-2a+9\right).

Tips and Tricks

  • When multiplying polynomials, always apply the distributive property.
  • Combine like terms to simplify the expression.
  • Use the distributive property to find the product of two or more polynomials.

Related Topics

  • Distributive Property
  • Combining Like Terms
  • Multiplying Polynomials

Final Answer

The final answer is D. 12a3+2a2+23a+1812a^3 + 2a^2 + 23a + 18.

Introduction

Multiplying polynomials is a fundamental concept in algebra that can be challenging for some students. In this article, we will answer some frequently asked questions (FAQs) about multiplying polynomials.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac. This means that we can multiply a single term by two or more terms and then add the results.

Q: How do I apply the distributive property when multiplying polynomials?

A: To apply the distributive property when multiplying polynomials, we need to multiply each term in the first polynomial by each term in the second polynomial. This means that we need to multiply each term in the first polynomial by each term in the second polynomial and then add the results.

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

A: Multiplying polynomials is similar to multiplying numbers, but with polynomials, we need to apply the distributive property. When multiplying numbers, we can simply multiply the numbers together, but when multiplying polynomials, we need to multiply each term in the first polynomial by each term in the second polynomial.

Q: How do I simplify the expression after multiplying polynomials?

A: To simplify the expression after multiplying polynomials, we need to combine like terms. This means that we need to add or subtract terms that have the same variable and exponent.

Q: What is the importance of multiplying polynomials?

A: Multiplying polynomials is an important concept in algebra because it allows us to simplify complex expressions. By multiplying polynomials, we can find the product of two or more polynomials and then simplify the expression.

Q: Can I use a calculator to multiply polynomials?

A: Yes, you can use a calculator to multiply polynomials. However, it's always a good idea to check your work by multiplying the polynomials by hand to make sure that you get the correct answer.

Q: What are some common mistakes to avoid when multiplying polynomials?

A: Some common mistakes to avoid when multiplying polynomials include:

  • Not applying the distributive property
  • Not combining like terms
  • Not checking your work
  • Not using the correct order of operations

Q: How can I practice multiplying polynomials?

A: You can practice multiplying polynomials by working through examples and exercises. You can also use online resources, such as math websites and apps, to practice multiplying polynomials.

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

A: Multiplying polynomials has many real-world applications, including:

  • Calculating the area and perimeter of shapes
  • Finding the volume of solids
  • Modeling population growth and decline
  • Analyzing data and making predictions

Conclusion

In conclusion, multiplying polynomials is an important concept in algebra that can be challenging for some students. By understanding the distributive property and how to apply it, we can simplify complex expressions and find the product of two or more polynomials. We hope that this article has helped to answer some of your frequently asked questions about multiplying polynomials.

Final Answer

The final answer is that multiplying polynomials is a fundamental concept in algebra that can be used to simplify complex expressions and find the product of two or more polynomials.