Questions: Multiply. 4 x^2(2 x^3-x^2-18 x+15)

Multiply. 4 x^2(2 x^3-x^2-18 x+15)
Transcript text: Multiply. \[ 4 x^{2}\left(2 x^{3}-x^{2}-18 x+15\right) \]
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Solution

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Solution Steps

To multiply the polynomial \(4x^2\) by the polynomial \(2x^3 - x^2 - 18x + 15\), distribute \(4x^2\) to each term inside the parentheses. This involves multiplying \(4x^2\) by each term in the polynomial separately and then combining the results.

Step 1: Distribute \(4x^2\) to Each Term in the Polynomial

To multiply the polynomial \(4x^2\) by the polynomial \(2x^3 - x^2 - 18x + 15\), distribute \(4x^2\) to each term inside the parentheses:

\[ 4x^2 \cdot (2x^3 - x^2 - 18x + 15) = 4x^2 \cdot 2x^3 + 4x^2 \cdot (-x^2) + 4x^2 \cdot (-18x) + 4x^2 \cdot 15 \]

Step 2: Perform the Multiplication

Calculate each term separately:

  • \(4x^2 \cdot 2x^3 = 8x^5\)
  • \(4x^2 \cdot (-x^2) = -4x^4\)
  • \(4x^2 \cdot (-18x) = -72x^3\)
  • \(4x^2 \cdot 15 = 60x^2\)
Step 3: Combine the Results

Combine all the terms to form the final polynomial:

\[ 8x^5 - 4x^4 - 72x^3 + 60x^2 \]

Final Answer

\(\boxed{8x^5 - 4x^4 - 72x^3 + 60x^2}\)

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