Find The Product. Simplify Your Answer.$\[(3c + 3)(3c^2 - 3c + 4)\\]

Introduction

When it comes to simplifying algebraic expressions, one of the most common operations is multiplying two or more polynomials together. In this article, we will focus on simplifying the product of two polynomials, specifically the expression $(3c + 3)(3c^2 - 3c + 4)$. We will break down the steps involved in simplifying this expression and provide a clear, step-by-step guide on how to do it.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at what we're dealing with. The expression $(3c + 3)(3c^2 - 3c + 4)$ is a product of two polynomials. The first polynomial is $3c + 3$, and the second polynomial is $3c^2 - 3c + 4$. Our goal is to simplify this expression by multiplying the two polynomials together.

Distributive Property

To simplify the expression, we will use the distributive property, which states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

We can apply this property to our expression by multiplying each term in the first polynomial by each term in the second polynomial.

Step 1: Multiply Each Term

To simplify the expression, we will multiply each term in the first polynomial by each term in the second polynomial. This will give us a total of four terms, which we will then combine.

$(3c + 3)(3c^2 - 3c + 4) = (3c)(3c^2) + (3c)(-3c) + (3c)(4) + (3)(3c^2) + (3)(-3c) + (3)(4)$

Step 2: Simplify Each Term

Now that we have multiplied each term, we can simplify each term by combining like terms.

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

$(3c)(-3c) = -9c^2$

$(3c)(4) = 12c$

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

$(3)(-3c) = -9c$

$(3)(4) = 12$

Step 3: Combine Like Terms

Now that we have simplified each term, we can combine like terms to get the final simplified expression.

$9c^3 - 9c^2 + 12c + 9c^2 - 9c + 12$

Step 4: Final Simplification

Finally, we can simplify the expression by combining like terms.

$9c^3 - 9c^2 + 12c + 9c^2 - 9c + 12 = 9c^3 + 12c - 9c + 12$

Step 5: Final Answer

After combining like terms, we get the final simplified expression.

$9c^3 + 3c + 12$

Conclusion

In this article, we have simplified the product of two polynomials, specifically the expression $(3c + 3)(3c^2 - 3c + 4)$. We have broken down the steps involved in simplifying this expression and provided a clear, step-by-step guide on how to do it. By following these steps, you should be able to simplify any product of two polynomials.

Frequently Asked Questions

• Q: What is the distributive property? A: The distributive property is a mathematical property that states that for any real numbers $a$, $b$, and $c$, the following equation holds: $a(b + c) = ab + ac$.
• Q: How do I simplify a product of two polynomials? A: To simplify a product of two polynomials, you can use the distributive property to multiply each term in the first polynomial by each term in the second polynomial, and then combine like terms.
• Q: What is the final simplified expression for the product of $(3c + 3)(3c^2 - 3c + 4)$? A: The final simplified expression for the product of $(3c + 3)(3c^2 - 3c + 4)$ is $9c^3 + 3c + 12$.

Additional Resources

• For more information on the distributive property, see Wikipedia.
• For more information on simplifying algebraic expressions, see Mathway.
• For more information on polynomials, see Khan Academy.

Final Thoughts

Simplifying algebraic expressions can be a challenging task, but with the right tools and techniques, it can be made much easier. By following the steps outlined in this article, you should be able to simplify any product of two polynomials. Remember to use the distributive property to multiply each term, and then combine like terms to get the final simplified expression. With practice and patience, you will become a master of simplifying algebraic expressions in no time.

Introduction

In our previous article, we simplified the product of two polynomials, specifically the expression $(3c + 3)(3c^2 - 3c + 4)$. We broke down the steps involved in simplifying this expression and provided a clear, step-by-step guide on how to do it. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers $a$, $b$, and $c$, the following equation holds: $a(b + c) = ab + ac$. This property allows us to multiply each term in one polynomial by each term in another polynomial.

Q: How do I simplify a product of two polynomials?

A: To simplify a product of two polynomials, you can use the distributive property to multiply each term in the first polynomial by each term in the second polynomial, and then combine like terms.

Q: What is the difference between a polynomial and an algebraic expression?

A: A polynomial is an expression that consists of variables and coefficients, and the variables are raised to non-negative integer powers. An algebraic expression, on the other hand, is a general term that refers to any expression that involves variables and coefficients.

Q: How do I know when to use the distributive property?

A: You should use the distributive property whenever you need to multiply two or more polynomials together. This property is essential for simplifying complex algebraic expressions.

Q: Can I simplify an expression that has more than two polynomials?

A: Yes, you can simplify an expression that has more than two polynomials. You can use the distributive property to multiply each polynomial by each other polynomial, and then combine like terms.

Q: What is the final simplified expression for the product of $(3c + 3)(3c^2 - 3c + 4)$?

A: The final simplified expression for the product of $(3c + 3)(3c^2 - 3c + 4)$ is $9c^3 + 3c + 12$.

Q: How do I know if an expression is already simplified?

A: An expression is already simplified if there are no like terms that can be combined. In other words, if the expression cannot be simplified further, it is already in its simplest form.

Q: Can I use the distributive property to simplify an expression that has variables with exponents?

A: Yes, you can use the distributive property to simplify an expression that has variables with exponents. However, you need to be careful when multiplying variables with exponents, as the exponent of the product is the sum of the exponents of the individual variables.

Q: How do I simplify an expression that has a negative coefficient?

A: To simplify an expression that has a negative coefficient, you can multiply the entire expression by -1 to make the coefficient positive. Then, you can use the distributive property to simplify the expression.

Conclusion

Simplifying algebraic expressions can be a challenging task, but with the right tools and techniques, it can be made much easier. By following the steps outlined in this article, you should be able to simplify any product of two polynomials. Remember to use the distributive property to multiply each term, and then combine like terms to get the final simplified expression. With practice and patience, you will become a master of simplifying algebraic expressions in no time.

Additional Resources

• For more information on the distributive property, see Wikipedia.
• For more information on simplifying algebraic expressions, see Mathway.
• For more information on polynomials, see Khan Academy.

Final Thoughts

Simplifying algebraic expressions is an essential skill that is used in many areas of mathematics and science. By mastering this skill, you will be able to solve complex problems and make sense of complex data. Remember to practice regularly and to seek help when you need it. With time and effort, you will become a master of simplifying algebraic expressions.