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Chapter 5: Problem 62

Multiply and check. $$ (4 x-5 x-3)\left(1+2 x^{2}\right) $$

Short Answer

Expert verified
The product is \( -2x^3 - 6x^2 - x - 3\).

Step by step solution

01

- Distribute each term

Expand the expression by distributing each term in \(4x - 5x - 3\) by every term in \(1 + 2x^2\). This means multiplying \(4x\) by \(1\) and \(2x^2\), \(-5x\) by \(1\) and \(2x^2\), and \(-3\) by \(1\) and \(2x^2\).
02

- Multiply each term

Calculate each individual multiplication term-by-term: \(4x \cdot 1 = 4x\), \(4x \cdot 2x^2 = 8x^3\), \(-5x \cdot 1 = -5x\), \(-5x \cdot 2x^2 = -10x^3\), \(-3 \cdot 1 = -3\), \(-3 \cdot 2x^2 = -6x^2\).
03

- Combine like terms

Combine the like terms together. Start by combining the cubic terms, then the linear terms: \(8x^3 - 10x^3 = -2x^3\), \(4x - 5x = -x\), and the constant and quadratic terms remain as they are because they don't have like terms: \(-6x^2\) and \(-3\).
04

- Write the simplified expression

Combine all of the simplified terms to get the final expression: \(-2x^3 - 6x^2 - x - 3\).

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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 crucial in polynomial multiplication. It allows us to distribute or 'spread' each term in one polynomial across the terms in another polynomial. In our exercise, we need to distribute each term in the polynomial \( 4x - 5x - 3 \) with every term in \( 1 + 2x^2 \).
This means we will individually multiply:
  • \( 4x \) by both \( 1 \) and \( 2x^2 \)
  • \(- 5x \) by both \( 1 \) and \( 2x^2 \)
  • \(- 3 \) by both \( 1 \) and \( 2x^2 \)

It鈥檚 like distributing groceries from one bag into several containers. Each item must go into each container, ensuring nothing is missed or left behind.
Combining Like Terms
After distributing, we often end up with terms that look similar. These are called 'like terms'.
Like terms are terms that have the same variable raised to the same power. In our solution, we identified the following terms from the multiplication:
  • \( 4x \) and \(-5x \)
  • \( 8x^3 \) and \(-10x^3 \)
  • \(-6x^2 \) and \(-3 \) (these have no like terms)

We then combine them:
  • \( 8x^3 - 10x^3 = -2x^3 \)
  • \( 4x - 5x = -x \)
  • \(-6x^2 \) and \(-3 \) stay the same

By combining like terms, we simplify the expression and make it easier to understand and work with.
Step-by-Step Solution
Here's a clear step-by-step solution for clarity.
Step 1: Distribute Each Term
Expand the expression by distributing:
  • \(4x(1) + 4x(2x^2) = 4x + 8x^3 \)
  • \(-5x(1) + -5x(2x^2) = -5x - 10x^3 \)
  • \(-3(1) + -3(2x^2) = -3 - 6x^2 \)

Step 2: Combine Like Terms
Identify and combine similar terms:
  • \(8x^3 - 10x^3 = -2x^3 \)
  • \(4x - 5x = -x \)
  • Consistent terms: \(-6x^2 \) and \(-3 \)

Step 3: Write the Final Expression
Piece it all together for the simplified expression:
\(-2x^3 - 6x^2 - x - 3 \).
By following these steps, we ensure no term is left out and the result is accurate.

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