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Answer :
Sure, let's multiply the polynomials [tex]\((4x^2 + 3x + 7)(8x - 5)\)[/tex] step by step.
1. Distribute each term in the first polynomial across each term in the second polynomial.
2. Multiply each term:
- Multiply [tex]\(4x^2\)[/tex] by both [tex]\(8x\)[/tex] and [tex]\(-5\)[/tex]:
- [tex]\(4x^2 \cdot 8x = 32x^3\)[/tex]
- [tex]\(4x^2 \cdot (-5) = -20x^2\)[/tex]
- Multiply [tex]\(3x\)[/tex] by both [tex]\(8x\)[/tex] and [tex]\(-5\)[/tex]:
- [tex]\(3x \cdot 8x = 24x^2\)[/tex]
- [tex]\(3x \cdot (-5) = -15x\)[/tex]
- Multiply [tex]\(7\)[/tex] by both [tex]\(8x\)[/tex] and [tex]\(-5\)[/tex]:
- [tex]\(7 \cdot 8x = 56x\)[/tex]
- [tex]\(7 \cdot (-5) = -35\)[/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-20x^2 + 24x^2 = 4x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-15x + 56x = 41x\)[/tex]
4. Write the final expression:
The product of the polynomials is [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex].
This matches option D.
1. Distribute each term in the first polynomial across each term in the second polynomial.
2. Multiply each term:
- Multiply [tex]\(4x^2\)[/tex] by both [tex]\(8x\)[/tex] and [tex]\(-5\)[/tex]:
- [tex]\(4x^2 \cdot 8x = 32x^3\)[/tex]
- [tex]\(4x^2 \cdot (-5) = -20x^2\)[/tex]
- Multiply [tex]\(3x\)[/tex] by both [tex]\(8x\)[/tex] and [tex]\(-5\)[/tex]:
- [tex]\(3x \cdot 8x = 24x^2\)[/tex]
- [tex]\(3x \cdot (-5) = -15x\)[/tex]
- Multiply [tex]\(7\)[/tex] by both [tex]\(8x\)[/tex] and [tex]\(-5\)[/tex]:
- [tex]\(7 \cdot 8x = 56x\)[/tex]
- [tex]\(7 \cdot (-5) = -35\)[/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-20x^2 + 24x^2 = 4x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-15x + 56x = 41x\)[/tex]
4. Write the final expression:
The product of the polynomials is [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex].
This matches option D.
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