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Answer :
To multiply the polynomials [tex]\((8x^2 + 6x + 8)\)[/tex] and [tex]\((6x - 5)\)[/tex], we will use the distributive property, which involves multiplying each term in the first polynomial by each term in the second polynomial, and then combining like terms.
Here is the step-by-step process:
1. Multiply each term in the first polynomial by each term in the second polynomial:
[tex]\((8x^2 + 6x + 8) \cdot (6x - 5)\)[/tex]
- Multiply [tex]\(8x^2\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- [tex]\(8x^2 \cdot 6x = 48x^3\)[/tex]
- [tex]\(8x^2 \cdot (-5) = -40x^2\)[/tex]
- Multiply [tex]\(6x\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- [tex]\(6x \cdot 6x = 36x^2\)[/tex]
- [tex]\(6x \cdot (-5) = -30x\)[/tex]
- Multiply [tex]\(8\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- [tex]\(8 \cdot 6x = 48x\)[/tex]
- [tex]\(8 \cdot (-5) = -40\)[/tex]
2. Collect and combine like terms:
- From the calculations above, the polynomial becomes:
[tex]\[
48x^3 + (-40x^2 + 36x^2) + (-30x + 48x) + (-40)
\][/tex]
- Combine like terms:
- [tex]\(48x^3\)[/tex] as the only [tex]\(x^3\)[/tex] term.
- For [tex]\(x^2\)[/tex], [tex]\(-40x^2 + 36x^2 = -4x^2\)[/tex].
- For [tex]\(x\)[/tex], [tex]\(-30x + 48x = 18x\)[/tex].
- The constant term is [tex]\(-40\)[/tex].
3. Write the final result:
The product of the polynomials is:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
Comparing this expression to the answer choices, we see that none of the options exactly match this result. Please ensure you have the correct versions of the answer choices, as sometimes there might be typographical errors or misprints. Based on calculations, your expanded polynomial is [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
Here is the step-by-step process:
1. Multiply each term in the first polynomial by each term in the second polynomial:
[tex]\((8x^2 + 6x + 8) \cdot (6x - 5)\)[/tex]
- Multiply [tex]\(8x^2\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- [tex]\(8x^2 \cdot 6x = 48x^3\)[/tex]
- [tex]\(8x^2 \cdot (-5) = -40x^2\)[/tex]
- Multiply [tex]\(6x\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- [tex]\(6x \cdot 6x = 36x^2\)[/tex]
- [tex]\(6x \cdot (-5) = -30x\)[/tex]
- Multiply [tex]\(8\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- [tex]\(8 \cdot 6x = 48x\)[/tex]
- [tex]\(8 \cdot (-5) = -40\)[/tex]
2. Collect and combine like terms:
- From the calculations above, the polynomial becomes:
[tex]\[
48x^3 + (-40x^2 + 36x^2) + (-30x + 48x) + (-40)
\][/tex]
- Combine like terms:
- [tex]\(48x^3\)[/tex] as the only [tex]\(x^3\)[/tex] term.
- For [tex]\(x^2\)[/tex], [tex]\(-40x^2 + 36x^2 = -4x^2\)[/tex].
- For [tex]\(x\)[/tex], [tex]\(-30x + 48x = 18x\)[/tex].
- The constant term is [tex]\(-40\)[/tex].
3. Write the final result:
The product of the polynomials is:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
Comparing this expression to the answer choices, we see that none of the options exactly match this result. Please ensure you have the correct versions of the answer choices, as sometimes there might be typographical errors or misprints. Based on calculations, your expanded polynomial is [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
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