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Answer :
Certainly! Let's multiply the polynomials [tex]\((4x^2 + 3x + 7)\)[/tex] and [tex]\((8x - 5)\)[/tex] step by step.
1. Distribute each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(32x^3\)[/tex].
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-20x^2\)[/tex].
2. Repeat the process for the second term in the first polynomial:
- Multiply [tex]\(3x\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(24x^2\)[/tex].
- Multiply [tex]\(3x\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-15x\)[/tex].
3. Finally, repeat the process for the last term in the first polynomial:
- Multiply [tex]\(7\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(56x\)[/tex].
- Multiply [tex]\(7\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-35\)[/tex].
4. Combine like terms:
- For [tex]\(x^3\)[/tex], the only term is [tex]\(32x^3\)[/tex].
- For [tex]\(x^2\)[/tex], combine [tex]\(-20x^2\)[/tex] and [tex]\(24x^2\)[/tex] to get [tex]\(4x^2\)[/tex].
- For [tex]\(x\)[/tex], combine [tex]\(-15x\)[/tex] and [tex]\(56x\)[/tex] to get [tex]\(41x\)[/tex].
- The constant term is [tex]\(-35\)[/tex].
Putting it all together, the polynomial becomes:
[tex]\[32x^3 + 4x^2 + 41x - 35\][/tex]
Based on the available choices, the correct answer is:
B. [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex]
1. Distribute each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(32x^3\)[/tex].
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-20x^2\)[/tex].
2. Repeat the process for the second term in the first polynomial:
- Multiply [tex]\(3x\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(24x^2\)[/tex].
- Multiply [tex]\(3x\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-15x\)[/tex].
3. Finally, repeat the process for the last term in the first polynomial:
- Multiply [tex]\(7\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(56x\)[/tex].
- Multiply [tex]\(7\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-35\)[/tex].
4. Combine like terms:
- For [tex]\(x^3\)[/tex], the only term is [tex]\(32x^3\)[/tex].
- For [tex]\(x^2\)[/tex], combine [tex]\(-20x^2\)[/tex] and [tex]\(24x^2\)[/tex] to get [tex]\(4x^2\)[/tex].
- For [tex]\(x\)[/tex], combine [tex]\(-15x\)[/tex] and [tex]\(56x\)[/tex] to get [tex]\(41x\)[/tex].
- The constant term is [tex]\(-35\)[/tex].
Putting it all together, the polynomial becomes:
[tex]\[32x^3 + 4x^2 + 41x - 35\][/tex]
Based on the available choices, the correct answer is:
B. [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex]
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