Answer :
Let's multiply the polynomials [tex]\((4x^2 + 3x + 7)(8x - 5)\)[/tex] step by step using the distributive property, also known as the distributive law of multiplication over addition. This means that each term of the first polynomial is multiplied by each term of the second polynomial.
1. Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\((8x - 5)\)[/tex]:
- [tex]\(4x^2 \cdot 8x = 32x^3\)[/tex]
- [tex]\(4x^2 \cdot (-5) = -20x^2\)[/tex]
2. Multiply [tex]\(3x\)[/tex] by each term in [tex]\((8x - 5)\)[/tex]:
- [tex]\(3x \cdot 8x = 24x^2\)[/tex]
- [tex]\(3x \cdot (-5) = -15x\)[/tex]
3. Multiply [tex]\(7\)[/tex] by each term in [tex]\((8x - 5)\)[/tex]:
- [tex]\(7 \cdot 8x = 56x\)[/tex]
- [tex]\(7 \cdot (-5) = -35\)[/tex]
4. Combine like terms:
- The [tex]\(x^3\)[/tex] term is [tex]\(32x^3\)[/tex].
- The [tex]\(x^2\)[/tex] terms are [tex]\(-20x^2\)[/tex] and [tex]\(24x^2\)[/tex]. Adding these gives [tex]\(4x^2\)[/tex].
- The [tex]\(x\)[/tex] terms are [tex]\(-15x\)[/tex] and [tex]\(56x\)[/tex]. Adding these gives [tex]\(41x\)[/tex].
- The constant term is [tex]\(-35\)[/tex].
Putting them all together, the polynomial is:
[tex]\[ 32x^3 + 4x^2 + 41x - 35 \][/tex]
Therefore, the answer is C. [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex].
1. Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\((8x - 5)\)[/tex]:
- [tex]\(4x^2 \cdot 8x = 32x^3\)[/tex]
- [tex]\(4x^2 \cdot (-5) = -20x^2\)[/tex]
2. Multiply [tex]\(3x\)[/tex] by each term in [tex]\((8x - 5)\)[/tex]:
- [tex]\(3x \cdot 8x = 24x^2\)[/tex]
- [tex]\(3x \cdot (-5) = -15x\)[/tex]
3. Multiply [tex]\(7\)[/tex] by each term in [tex]\((8x - 5)\)[/tex]:
- [tex]\(7 \cdot 8x = 56x\)[/tex]
- [tex]\(7 \cdot (-5) = -35\)[/tex]
4. Combine like terms:
- The [tex]\(x^3\)[/tex] term is [tex]\(32x^3\)[/tex].
- The [tex]\(x^2\)[/tex] terms are [tex]\(-20x^2\)[/tex] and [tex]\(24x^2\)[/tex]. Adding these gives [tex]\(4x^2\)[/tex].
- The [tex]\(x\)[/tex] terms are [tex]\(-15x\)[/tex] and [tex]\(56x\)[/tex]. Adding these gives [tex]\(41x\)[/tex].
- The constant term is [tex]\(-35\)[/tex].
Putting them all together, the polynomial is:
[tex]\[ 32x^3 + 4x^2 + 41x - 35 \][/tex]
Therefore, the answer is C. [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex].
