Question
What is \( (3 x-2) \) multiplied by \( \left(x^{2}+2 x+5\right) \)
Ask by Edwards Mckenzie. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The product is \( 3x^3 + 4x^2 + 11x - 10 \).
Solution
To multiply (3x – 2) by (x² + 2x + 5), use the distributive property:
1. Multiply 3x by each term in the second polynomial:
• 3x · x² = 3x³
• 3x · 2x = 6x²
• 3x · 5 = 15x
2. Multiply –2 by each term in the second polynomial:
• –2 · x² = –2x²
• –2 · 2x = –4x
• –2 · 5 = –10
3. Combine like terms:
3x³ + (6x² – 2x²) + (15x – 4x) – 10
= 3x³ + 4x² + 11x – 10
So, the product is 3x³ + 4x² + 11x – 10.
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Simplify this solution Beyond the Answer
To multiply \( (3x - 2) \) by \( (x^2 + 2x + 5) \), we use the distributive property, also known as the FOIL method for binomials. First, distribute \( 3x \): - \( 3x \cdot x^2 = 3x^3 \) - \( 3x \cdot 2x = 6x^2 \) - \( 3x \cdot 5 = 15x \) Next, distribute \( -2 \): - \( -2 \cdot x^2 = -2x^2 \) - \( -2 \cdot 2x = -4x \) - \( -2 \cdot 5 = -10 \) Now, combine all the results: \[ 3x^3 + 6x^2 + 15x - 2x^2 - 4x - 10 \] Combine like terms: \[ 3x^3 + (6x^2 - 2x^2) + (15x - 4x) - 10 \] This simplifies to: \[ 3x^3 + 4x^2 + 11x - 10 \] So, the final result is: \[ 3x^3 + 4x^2 + 11x - 10 \]
