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Chapter 4: Problem 74

Find each product. $$ 2 x^{2}(2 x-5)(x+3) $$

Short Answer

Expert verified
4x^4 + 2x^3 - 30x^2

Step by step solution

01

Expand the Expression Inside the Parentheses

First, expand the product \(2x - 5\)(x + 3)Use the distributive property:\[ (2x - 5)(x + 3) = 2x(x) + 2x(3) - 5(x) - 5(3) \]Simplify:\[ = 2x^2 + 6x - 5x - 15 \]Combine like terms:\[ = 2x^2 + x - 15 \]
02

Multiply by the Outside Term

Now multiply the expanded expression \(2x^2 + x - 15\) by \(2x^2\)Use the distributive property again:\[ 2x^2(2x^2 + x - 15) = 2x^2(2x^2) + 2x^2(x) + 2x^2(-15) \]Simplify each term:\[ = 4x^4 + 2x^3 - 30x^2 \]

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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 an essential rule in algebra that makes multiplying and simplifying expressions easier. It helps break down complex expressions into manageable parts.

When you encounter a problem like \(a(b + c)\), the distributive property allows you to distribute the \(a\) across the terms inside the parentheses: \[ a(b + c) = ab + ac \]
This means you multiply each term inside the parentheses by the term outside.

Let's look at an example: \[ (2x - 5)(x + 3) \]
Apply the distributive property to expand:
  • First, multiply \(2x\) with \(x\), giving \(2x^2\)
  • Next, multiply \(2x\) with \(3\), giving \(6x\)
  • Then, multiply \(-5 \) with \(x\), giving \ -5x \
  • Finally, multiply \(-5 \) with \(3\), giving \ -15 \


This results in: \[ 2x^2 + 6x - 5x - 15 \]

The distributive property simplifies complex expressions and prepares them for further steps.
combining like terms
Combining like terms is the next crucial step after using the distributive property. It involves grouping and simplifying expressions by merging terms with the same variables and exponents.

Let's start from the expanded expression: \[2x^2 + 6x - 5x - 15 \]
Here, \ 6x \ and \ -5x \ have the same variable part (both are x terms).

To combine these like terms: \[ 6x - 5x = x \]
Now our expression becomes:
  • \(2x^2\)
  • \(x\)
  • \(-15\)

This makes the simplified expression: \[ 2x^2 + x - 15 \]

Combining like terms helps clean up the expression, making it clearer and more manageable for further steps or evaluations.
simplification
Simplification is the final process of refining an expression to its simplest form. It usually follows using the distributive property and combining like terms.

After combining like terms in the previous steps, we reached: \[ 2x^2 + x - 15 \]
Next, we need to multiply by the outside term. In our case: \[ 2x^2 (2x^2 + x - 15) \]

Let's distribute \(2x^2\) across each term inside the parentheses:
  • \(2x^2 * 2x^2 = 4x^4\)
  • \(2x^2 * x = 2x^3\)
  • \(2x^2 * (-15) = -30x^2\)

This gives us: \[ 4x^4 + 2x^3 - 30x^2 \]
You've now simplified the expression fully!

This process demonstrates how simplification ties all the steps together鈥攆rom distributing terms, combining like terms, and reducing to the simplest form.

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