Bot GPT 4-o mini To find the product of the polynomials \( (2x^3 + 3x^2)(4x^4 - 5x^3 - 6x^2) \), we can use the distributive property (also known as the FOIL method for binomials, but we will apply it to this situation more generally).
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Distribute \( 2x^3 \) across \( (4x^4 - 5x^3 - 6x^2) \):
- \( 2x^3 \cdot 4x^4 = 8x^7 \)
- \( 2x^3 \cdot (-5x^3) = -10x^6 \)
- \( 2x^3 \cdot (-6x^2) = -12x^5 \)
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Distribute \( 3x^2 \) across \( (4x^4 - 5x^3 - 6x^2) \):
- \( 3x^2 \cdot 4x^4 = 12x^6 \)
- \( 3x^2 \cdot (-5x^3) = -15x^5 \)
- \( 3x^2 \cdot (-6x^2) = -18x^4 \)
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Combine all terms:
- From \( 2x^3 \): \( 8x^7 - 10x^6 - 12x^5 \)
- From \( 3x^2 \): \( 12x^6 - 15x^5 - 18x^4 \)
So we combine like terms:
- The \( x^7 \) term: \( 8x^7 \)
- The \( x^6 \) terms: \( -10x^6 + 12x^6 = 2x^6 \)
- The \( x^5 \) terms: \( -12x^5 - 15x^5 = -27x^5 \)
- The \( x^4 \) term: \( -18x^4 \)
Putting it all together, we get the product: \[ 8x^7 + 2x^6 - 27x^5 - 18x^4 \]
Now, let's compare with the options given:
- A: \( 8x^7 + 2x^6 - 27x^5 - 18x^4 \) (matches)
- B: \( 8x^7 - 22x^6 - 27x^5 - 18x^4 \)
- C: \( 8x^7 + 2x^6 - 3x^5 - 18x^4 \)
- D: \( 8x^7 + 22x^6 - 27x^5 + 18x^4 \)
The correct answer is A: \( 8x^7 + 2x^6 - 27x^5 - 18x^4 \).
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