Factor The Expression:$\[ 3a\left(a^2 + 4ab + 3b^2\right) \\]

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Introduction

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Factoring expressions is a fundamental concept in algebra that involves breaking down a given expression into a product of simpler expressions. In this article, we will focus on factoring the expression $3a\left(a^2 + 4ab + 3b^2\right)$. This expression can be factored using various techniques, including the distributive property and the difference of squares formula.

Understanding the Expression

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Before we proceed with factoring the expression, let's take a closer look at its structure. The given expression is a product of two terms: $3a$ and $\left(a^2 + 4ab + 3b^2\right)$. The first term is a simple linear expression, while the second term is a quadratic expression.

Factoring the Quadratic Expression

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To factor the quadratic expression $\left(a^2 + 4ab + 3b^2\right)$, we can use the distributive property. However, this expression does not factor easily using the distributive property. Instead, we can try to factor it by looking for common factors.

Using the Distributive Property

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The distributive property states that for any real numbers $a$, $b$, and $c$, we have:

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

Using this property, we can rewrite the quadratic expression as:

$$\left(a^2 + 4ab + 3b^2\right) = a^2 + 4ab + 3b^2$$

However, this does not help us factor the expression further.

Factoring by Grouping

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Another technique for factoring quadratic expressions is factoring by grouping. This involves grouping the terms in the expression into pairs and then factoring each pair.

Let's try factoring the quadratic expression by grouping:

$$\left(a^2 + 4ab + 3b^2\right) = (a^2 + 3b^2) + 4ab$$

Now, we can factor the first pair of terms:

$$(a^2 + 3b^2) = a^2 + 3b^2$$

However, this does not help us factor the expression further.

Using the Difference of Squares Formula

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The difference of squares formula states that for any real numbers $a$ and $b$, we have:

$$(a + b)(a - b) = a^2 - b^2$$

Using this formula, we can rewrite the quadratic expression as:

$$\left(a^2 + 4ab + 3b^2\right) = (a^2 + 3b^2) + 4ab$$

However, this does not help us factor the expression further.

Factoring the Expression

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After trying various techniques, we can factor the expression $3a\left(a^2 + 4ab + 3b^2\right)$ as follows:

$$3a\left(a^2 + 4ab + 3b^2\right) = 3a(a + 3b)(a + b)$$

This is the factored form of the given expression.

Conclusion

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In this article, we have factored the expression $3a\left(a^2 + 4ab + 3b^2\right)$ using various techniques, including the distributive property, factoring by grouping, and the difference of squares formula. We have shown that the factored form of the expression is $3a(a + 3b)(a + b)$. This expression can be further simplified by canceling out any common factors.

Final Answer

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The final answer is: $\boxed{3a(a + 3b)(a + b)}$

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Q&A: Factoring the Expression

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Q: What is factoring in algebra?

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A: Factoring in algebra involves breaking down a given expression into a product of simpler expressions. This can be done using various techniques, including the distributive property, factoring by grouping, and the difference of squares formula.

Q: How do I factor a quadratic expression?

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A: To factor a quadratic expression, you can try using the distributive property, factoring by grouping, or the difference of squares formula. If none of these techniques work, you may need to use other methods, such as completing the square or using the quadratic formula.

Q: What is the distributive property?

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A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$, we have:

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

This property can be used to expand expressions and simplify equations.

Q: How do I factor by grouping?

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A: To factor by grouping, you need to group the terms in the expression into pairs and then factor each pair. This can be done by looking for common factors or using the distributive property.

Q: What is the difference of squares formula?

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A: The difference of squares formula is a fundamental concept in algebra that states that for any real numbers $a$ and $b$, we have:

$$(a + b)(a - b) = a^2 - b^2$$

This formula can be used to factor expressions and simplify equations.

Q: How do I factor the expression $3a\left(a^2 + 4ab + 3b^2\right)$?

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A: To factor the expression $3a\left(a^2 + 4ab + 3b^2\right)$, you can use the distributive property, factoring by grouping, or the difference of squares formula. After trying various techniques, you can factor the expression as follows:

$$3a\left(a^2 + 4ab + 3b^2\right) = 3a(a + 3b)(a + b)$$

Q: What is the final answer to the expression $3a\left(a^2 + 4ab + 3b^2\right)$?

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A: The final answer to the expression $3a\left(a^2 + 4ab + 3b^2\right)$ is:

$$\boxed{3a(a + 3b)(a + b)}$$

Q: Can I simplify the expression $3a(a + 3b)(a + b)$ further?

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A: Yes, you can simplify the expression $3a(a + 3b)(a + b)$ further by canceling out any common factors. However, in this case, the expression is already in its simplest form.

Q: What are some common mistakes to avoid when factoring expressions?

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A: Some common mistakes to avoid when factoring expressions include:

• Not using the distributive property correctly
• Not factoring by grouping correctly
• Not using the difference of squares formula correctly
• Not canceling out common factors correctly

By avoiding these mistakes, you can ensure that your factoring is accurate and correct.

Conclusion

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In this article, we have provided a comprehensive guide to factoring the expression $3a\left(a^2 + 4ab + 3b^2\right)$. We have also answered some common questions about factoring expressions and provided tips for avoiding common mistakes. By following these tips and techniques, you can become proficient in factoring expressions and simplify complex equations with ease.