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Chapter 5: Problem 15

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

Short Answer

Expert verified
-6a^4 + 4a^3 - 6a^2

Step by step solution

01

Write Down the Expression

We start with the expression given in the problem: \(-2 a^{2} (3 a^{2} - 2 a + 3)\).
02

Distribute the First Term

We apply the distributive property, multiplying \(-2 a^{2}\) by each term inside the parentheses: \(-2 a^{2} imes 3 a^{2} = -6 a^{4}\).
03

Distribute the Second Term

Continuing with distribution, multiply \(-2 a^{2}\) by the second term inside the parentheses:\(-2 a^{2} imes (-2 a) = 4 a^{3}\).
04

Distribute the Third Term

Finally, multiply \(-2 a^{2}\) by the third term inside the parentheses: \(-2 a^{2} imes 3 = -6 a^{2}\).
05

Write the Simplified Expression

Combine the results from Steps 2, 3, and 4 into a single expression: \(-6 a^{4} + 4 a^{3} - 6 a^{2}\).

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Key Concepts

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

Distributive Property
The distributive property is a mathematical rule that can make algebraic operations easier and more streamlined. When we use this property, we're essentially multiplying a single term by each of the terms inside a set of parentheses. This property can be summarized as:
  • If you have an expression like this: \( a(b + c) \), using the distributive property, it becomes: \( ab + ac \).
In the context of polynomial multiplication, like our given problem, the distributive property helps in multiplying monomials with polynomials. Each term outside the parentheses is multiplied individually by every term inside the parentheses. This is repeated until every combination is accounted for.
For our exercise,
  • We multiplied \( -2a^2 \) by each term inside the parentheses \( (3a^2 - 2a + 3) \).
This step-by-step multiplication builds the foundation for our expression.
Algebraic Expressions
Algebraic expressions are like the language of algebra. They consist of numbers, variables, and operations. Each expression acts like a mathematical sentence that can tell us how numbers and variables relate to each other.
In algebraic expressions:
  • Variables stand for unknown values. They鈥檙e often represented by letters like \( a, b, ext{or} x \).
  • Constants are known values, such as numbers like 2 or 3.
  • Coefficients are numbers that multiply the variables.
In our exercise, \(-2a^2(3a^2 - 2a + 3)\), we deal with various components:
  • The coefficient is \(-2\), multiplying the variable \(a^2\).
  • Inside the parentheses, each term is its own algebraic expression: \(3a^2, -2a, ext{and} 3\).
The beauty of algebra lies in combining these elements to solve and simplify problems.
Order of Operations
Order of operations is a fundamental principle in mathematics ensuring that expressions are evaluated consistently and correctly. This concept is often remembered with the acronym PEMDAS:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
When handling algebraic expressions, we need to apply these rules appropriately. This is crucial, especially in more complex equations, to avoid errors and ensure the correct outcome.
In our original problem, the order of operations guided us:
  • First, we addressed the operations within the parentheses by using the distributive property.
  • We didn't have any exponents to solve separately, but they are evident in the terms themselves such as \(a^2\).
  • Subsequent multiplication involved handling the distribution across each term inside the resulting algebraic expression.
By adhering to the order of operations, we make sure that our multiplication and combinations yield the right, simplified answer.

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Most popular questions from this chapter

The perimeter of a square is \(\left(12 x^{3}+4 x-16\right)\) feet. Find the length of its side.

Perform the indicated operation. $$(5 x)^{2}$$

Perform the indicated operation. $$\left(-3 y^{3}\right)^{2}$$

Evaluate each expression using exponential rules. Write each result in standard notation. See Example 7. $$ \frac{1.4 \times 10^{-2}}{7 \times 10^{-8}} $$

Match each expression on the left to the equivalent expression on the right. See the Concept Check in this section. $$ (a+b)^{2}(a-b)^{2} $$ $$A. a^{2}-b^{2}$$ $$B. a^{2}+b^{2}$$ $$C. a^{2}-2 a b+b^{2}$$ $$D. a^{2}+2 a b+b^{2}$$ $$E. none of these$$
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