Problem 12 Find each product. $$ -3 a^{... [FREE SOLUTION]

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Chapter 5: Problem 12

Find each product. $$ -3 a^{4}(4+a) $$

Short Answer

Expert verified

$ -12a^{4} - 3a^{5} $

Step by step solution

Distribute $ -3a^{4} $ to each term inside the parentheses

Multiply $ -3a^{4} $ by each term inside the parentheses. The terms are $ 4 $ and $ a $.

Multiply $ -3a^{4} $ by $ 4 $

Calculate $ -3a^{4} \cdot \ 4 $, which simplifies to $ -12a^{4} $.

Multiply $ -3a^{4} $ by $ a $

Calculate $ -3a^{4} \cdot \ a $, which simplifies to $ -3a^{5} $.

Combine the terms

Combine the results from previous steps: $ -12a^{4} $ and $ -3a^{5} $. This gives $ -12a^{4} - 3a^{5} $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Distributive Property

The distributive property is a fundamental principle in algebra, essential for polynomial multiplication. It allows us to multiply a single term by every term within a parenthesis. This process ensures each term is correctly accounted for in the final product. The distributive property is often described as:

\[ a(b + c) = ab + ac \]

Applying this property to our example, we start by distributing $-3a^4 $ to each term inside the parentheses $(4 + a) $. This means:

Multiply $-3a^4 $ by \4\
Multiply $-3a^4 $ by \a \

This ensures that every term in the parentheses is effectively taken into account, enabling us to proceed with the multiplication correctly.

Applying Exponent Rules

Exponent rules are crucial for simplifying terms after applying the distributive property. Let's break it down:

Multiplication of constants: When multiplying terms like $-3a^4 $ and \4 \, you handle the coefficients (numerical parts) separately. Thus, \[ -3 \times 4 = -12 \].
Multiplication of variables with exponents: When multiplying $-3a^4 $ and \a \, ensure you correctly apply the exponent rules. Here, you add the exponents of the variables:

\[ a^4 \times a = a^{4+1} = a^5 \].

So, by following the exponent rules:

$-3a^4 \times a = -3a^5 $. Combining this with our earlier result gives us our terms: $-12a^4 $ and $-3a^5 $.

Combining Like Terms

Once you have distributed and simplified all terms using their exponents, the last step is to combine like terms.

In polynomial expressions, like terms have the same variable raised to the same power. Here, $-12a^4 $ and $-3a^5 $ are not like terms.

The term $-12a^4 $ has \a \ raised to the 4th power.
The term $-3a^5 $ has \a \ raised to the 5th power.

Therefore, these terms can't be combined any further, and the solutions remain separate:

$-12a^4 - 3a^5 $.
This final step ensures your solution is fully simplified and expressed correctly.

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91影视

Find each product. $$ -3 a^{4}(4+a) $$

Short Answer

Step by step solution

Distribute \( -3a^{4} \) to each term inside the parentheses

Multiply \( -3a^{4} \) by \( 4 \)

Multiply \( -3a^{4} \) by \( a \)

Combine the terms

Key Concepts

Understanding the Distributive Property

Applying Exponent Rules

Combining Like Terms

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