Problem 31 Multiply. $$-a\left(-2 a^{2}-3 a... [FREE SOLUTION]

Chapter 9: Problem 31

Multiply. $$-a\left(-2 a^{2}-3 a-2\right)$$

Short Answer

Expert verified

The result of the multiplication is $2a^{3} + 3a^{2} + 2a$

Step by step solution

Distribute the Negative Term

Distribute $-a$ across each term in the parentheses $-2 a^{2}-3 a-2$. This results in $a * 2a^{2}$, $a * 3a$, and $a * 2$

Simplify the Terms

Next, simplify $a * 2a^{2}$ to $2a^{3}$, simplify $a * 3a$ to $3a^{2}$, and simplify $a * 2$ to $2a$

Write the Final Answer

Now, combine all the terms which results in the final expression $2a^{3} + 3a^{2} + 2a$

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Key Concepts

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

Distribution in Algebra

The concept of distribution in algebra refers to the process of multiplying a single term across the terms inside parentheses. It's a critical technique used to simplify expressions and solve equations. In the given exercise, we have the expression \[-a(-2a^2-3a-2)\]We need to distribute $-a$ through each term inside the parentheses. Here鈥檚 how this works:

First, multiply $-a$ with $-2a^2$, resulting in $2a^3$. Keep in mind that multiplying two negatives gives a positive result.
Next, distribute $-a$ to $-3a$, which results in $3a^2$.
Finally, multiply $-a$ by $-2$, yielding $2a$. Again, two negatives make a positive.

By following this process, you ensure that each term in the parentheses is multiplied correctly by the term outside. Using distribution effectively sets the stage for simplifying complex algebraic expressions.

Simplifying Expressions

Simplifying expressions involves reducing them into their simplest form. This helps in better understanding and solving algebraic equations. Once we have distributed $-a$ across the terms inside the parentheses, we need to simplify each resulting term. Here's how:

The product $a \times 2a^2$ simplifies to $2a^3$. You multiply the coefficients and add the exponents of like bases.
For the term $a \times 3a$, the simplification results in $3a^2$.
Lastly, $a \times 2$ simplifies to $2a$.

Now that we've simplified each term, we combine them to write the expression in its simplest form. Simplifying helps ensure that mathematical expressions are more manageable and easier to work with, especially in solving equations or making calculations.

Monomials

Monomials are one-piece expressions consisting of numbers, variables, or the product of numbers and variables, where variables are raised to a power. Recognizing monomials is fundamental in distribution and simplification processes. In the provided exercise, each term we deal with after distribution is a monomial.

$2a^3$ is a monomial where the coefficient (the number) is 2, and the variable part is $a^3$. A monomial can have multiple variables, but in this case, we only have one.
Similarly, $3a^2$ is another monomial, with 3 as the coefficient and $a^2$ as the variable part.
Lastly, $2a$ is also a monomial where the coefficient is 2 and the variable is $a$.

Understanding monomials helps in operations like addition, subtraction, and multiplication of algebraic expressions, making it easier to follow steps such as distribution and simplification.

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91影视

Multiply. $$-a\left(-2 a^{2}-3 a-2\right)$$

Short Answer

Step by step solution

Distribute the Negative Term

Simplify the Terms

Write the Final Answer

Key Concepts

Distribution in Algebra

Simplifying Expressions

Monomials

One App. One Place for Learning.

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