Simplify The Expression:$3x^3 - 24x^2y + 48xy^2$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the techniques involved in simplifying expressions with multiple terms. In this article, we will focus on simplifying the given expression: $3x^3 - 24x^2y + 48xy^2$. We will use various techniques, including factoring and combining like terms, to simplify the expression.

Understanding the Expression

The given expression is a polynomial expression with three terms: $3x^3$, $-24x^2y$, and $48xy^2$. Each term has a variable part and a coefficient. The variable part is the part of the term that contains the variables, and the coefficient is the numerical part of the term.

Identifying Like Terms

To simplify the expression, we need to identify the like terms. Like terms are terms that have the same variable part. In this expression, the like terms are $3x^3$ and $-24x^2y$ do not have the same variable part, but $-24x^2y$ and $48xy^2$ do not have the same variable part either, however $-24x^2y$ and $48xy^2$ can be simplified by factoring out a common factor of $-4xy$.

Factoring Out a Common Factor

We can factor out a common factor of $-4xy$ from the terms $-24x^2y$ and $48xy^2$. This will give us:

$$-24x^2y = -4xy(6x)$$

$$48xy^2 = -4xy(12y)$$

Now, we can rewrite the expression as:

$$3x^3 - 4xy(6x + 12y)$$

Simplifying the Expression

We can simplify the expression further by combining the like terms. The like terms are $3x^3$ and $-4xy(6x + 12y)$. We can combine these terms by adding or subtracting their coefficients.

Combining Like Terms

To combine the like terms, we need to add or subtract their coefficients. The coefficient of $3x^3$ is 3, and the coefficient of $-4xy(6x + 12y)$ is -4. We can combine these terms by adding or subtracting their coefficients:

$$3x^3 - 4xy(6x + 12y) = 3x^3 - 24x^2y + 48xy^2$$

However, we can simplify the expression further by factoring out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$.

Factoring Out a Common Factor

We can factor out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$. This will give us:

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

$$-24x^2y = 3x^2(-8y)$$

Now, we can rewrite the expression as:

$$3x^2(x - 8y) - 48xy^2$$

Simplifying the Expression

We can simplify the expression further by combining the like terms. The like terms are $3x^2(x - 8y)$ and $-48xy^2$. We can combine these terms by adding or subtracting their coefficients.

Combining Like Terms

To combine the like terms, we need to add or subtract their coefficients. The coefficient of $3x^2(x - 8y)$ is 3, and the coefficient of $-48xy^2$ is -48. We can combine these terms by adding or subtracting their coefficients:

$$3x^2(x - 8y) - 48xy^2 = 3x^3 - 24x^2y + 48xy^2$$

However, we can simplify the expression further by factoring out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$.

Factoring Out a Common Factor

We can factor out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$. This will give us:

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

$$-24x^2y = 3x^2(-8y)$$

Now, we can rewrite the expression as:

$$3x^2(x - 8y) - 48xy^2$$

Simplifying the Expression

We can simplify the expression further by combining the like terms. The like terms are $3x^2(x - 8y)$ and $-48xy^2$. We can combine these terms by adding or subtracting their coefficients.

Combining Like Terms

To combine the like terms, we need to add or subtract their coefficients. The coefficient of $3x^2(x - 8y)$ is 3, and the coefficient of $-48xy^2$ is -48. We can combine these terms by adding or subtracting their coefficients:

$$3x^2(x - 8y) - 48xy^2 = 3x^3 - 24x^2y + 48xy^2$$

However, we can simplify the expression further by factoring out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$.

Factoring Out a Common Factor

We can factor out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$. This will give us:

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

$$-24x^2y = 3x^2(-8y)$$

Now, we can rewrite the expression as:

$$3x^2(x - 8y) - 48xy^2$$

Simplifying the Expression

We can simplify the expression further by combining the like terms. The like terms are $3x^2(x - 8y)$ and $-48xy^2$. We can combine these terms by adding or subtracting their coefficients.

Combining Like Terms

To combine the like terms, we need to add or subtract their coefficients. The coefficient of $3x^2(x - 8y)$ is 3, and the coefficient of $-48xy^2$ is -48. We can combine these terms by adding or subtracting their coefficients:

$$3x^2(x - 8y) - 48xy^2 = 3x^3 - 24x^2y + 48xy^2$$

However, we can simplify the expression further by factoring out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$.

Factoring Out a Common Factor

We can factor out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$. This will give us:

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

$$-24x^2y = 3x^2(-8y)$$

Now, we can rewrite the expression as:

$$3x^2(x - 8y) - 48xy^2$$

Simplifying the Expression

We can simplify the expression further by combining the like terms. The like terms are $3x^2(x - 8y)$ and $-48xy^2$. We can combine these terms by adding or subtracting their coefficients.

Combining Like Terms

To combine the like terms, we need to add or subtract their coefficients. The coefficient of $3x^2(x - 8y)$ is 3, and the coefficient of $-48xy^2$ is -48. We can combine these terms by adding or subtracting their coefficients:

$$3x^2(x - 8y) - 48xy^2 = 3x^3 - 24x^2y + 48xy^2$$

However, we can simplify the expression further by factoring out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$.

Factoring Out a Common Factor

We can factor out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$. This will give us:

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

$$-24x^2y = 3x^2(-8y)$$

Now, we can rewrite the expression as:

$$3x^2(x - 8y) - 48xy^2$$

Conclusion

In this article, we simplified the expression $3x^3 - 24x^2y + 48xy^2$ by factoring out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$. We then combined the like terms to simplify the expression further. The final simplified expression is $3x^2(x - 8y) - 48xy^2$.

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Introduction

In our previous article, we simplified the expression $3x^3 - 24x^2y + 48xy^2$ by factoring out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$. We then combined the like terms to simplify the expression further. In this article, we will answer some frequently asked questions about simplifying the expression.

Q&A

Q: What is the first step in simplifying the expression $3x^3 - 24x^2y + 48xy^2$?

A: The first step in simplifying the expression $3x^3 - 24x^2y + 48xy^2$ is to identify the like terms. Like terms are terms that have the same variable part.

Q: How do I identify the like terms in the expression $3x^3 - 24x^2y + 48xy^2$?

A: To identify the like terms in the expression $3x^3 - 24x^2y + 48xy^2$, we need to look for terms that have the same variable part. In this expression, the like terms are $3x^3$ and $-24x^2y$ do not have the same variable part, but $-24x^2y$ and $48xy^2$ do not have the same variable part either, however $-24x^2y$ and $48xy^2$ can be simplified by factoring out a common factor of $-4xy$.

Q: What is the next step in simplifying the expression $3x^3 - 24x^2y + 48xy^2$?

A: The next step in simplifying the expression $3x^3 - 24x^2y + 48xy^2$ is to factor out a common factor from the like terms. In this case, we can factor out a common factor of $-4xy$ from the terms $-24x^2y$ and $48xy^2$.

Q: How do I factor out a common factor from the like terms in the expression $3x^3 - 24x^2y + 48xy^2$?

A: To factor out a common factor from the like terms in the expression $3x^3 - 24x^2y + 48xy^2$, we need to identify the common factor and divide each term by that factor. In this case, the common factor is $-4xy$, so we can factor it out as follows:

$$-24x^2y = -4xy(6x)$$

$$48xy^2 = -4xy(12y)$$

Q: What is the final simplified expression of $3x^3 - 24x^2y + 48xy^2$?

A: The final simplified expression of $3x^3 - 24x^2y + 48xy^2$ is $3x^2(x - 8y) - 48xy^2$.

Q: How do I combine the like terms in the expression $3x^2(x - 8y) - 48xy^2$?

A: To combine the like terms in the expression $3x^2(x - 8y) - 48xy^2$, we need to add or subtract their coefficients. In this case, the coefficient of $3x^2(x - 8y)$ is 3, and the coefficient of $-48xy^2$ is -48. We can combine these terms by adding or subtracting their coefficients:

$$3x^2(x - 8y) - 48xy^2 = 3x^3 - 24x^2y + 48xy^2$$

However, we can simplify the expression further by factoring out a common factor of $3x^2$ from the terms $3x^3$ and $-24x^2y$.

Conclusion

In this article, we answered some frequently asked questions about simplifying the expression $3x^3 - 24x^2y + 48xy^2$. We covered topics such as identifying like terms, factoring out a common factor, and combining like terms. We hope that this article has been helpful in understanding the process of simplifying the expression $3x^3 - 24x^2y + 48xy^2$.