Answer :
To multiply the polynomials [tex]\((8x^2 + 6x + 8)(6x - 5)\)[/tex], we can apply the distributive property, also known as the FOIL (First, Outer, Inner, Last) method, which is commonly used for multiplying binomials, but can also extend to larger polynomials. However, here is a step-by-step breakdown of how to multiply every term of the first polynomial by every term of the second polynomial:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\(8x^2\)[/tex] by [tex]\(6x\)[/tex] to get [tex]\(48x^3\)[/tex].
- Multiply [tex]\(8x^2\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-40x^2\)[/tex].
- Multiply [tex]\(6x\)[/tex] by [tex]\(6x\)[/tex] to get [tex]\(36x^2\)[/tex].
- Multiply [tex]\(6x\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-30x\)[/tex].
- Multiply [tex]\(8\)[/tex] by [tex]\(6x\)[/tex] to get [tex]\(48x\)[/tex].
- Multiply [tex]\(8\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-40\)[/tex].
2. Combine like terms:
- The terms involving [tex]\(x^3\)[/tex]:
- There is only one term: [tex]\(48x^3\)[/tex].
- The terms involving [tex]\(x^2\)[/tex]:
- Combine [tex]\(-40x^2\)[/tex] and [tex]\(36x^2\)[/tex] to get [tex]\(-4x^2\)[/tex].
- The terms involving [tex]\(x\)[/tex]:
- Combine [tex]\(-30x\)[/tex] and [tex]\(48x\)[/tex] to get [tex]\(18x\)[/tex].
- The constant terms:
- The only constant is [tex]\(-40\)[/tex].
3. Write the result as a single polynomial:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
The correct answer is [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
This matches option B: [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\(8x^2\)[/tex] by [tex]\(6x\)[/tex] to get [tex]\(48x^3\)[/tex].
- Multiply [tex]\(8x^2\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-40x^2\)[/tex].
- Multiply [tex]\(6x\)[/tex] by [tex]\(6x\)[/tex] to get [tex]\(36x^2\)[/tex].
- Multiply [tex]\(6x\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-30x\)[/tex].
- Multiply [tex]\(8\)[/tex] by [tex]\(6x\)[/tex] to get [tex]\(48x\)[/tex].
- Multiply [tex]\(8\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-40\)[/tex].
2. Combine like terms:
- The terms involving [tex]\(x^3\)[/tex]:
- There is only one term: [tex]\(48x^3\)[/tex].
- The terms involving [tex]\(x^2\)[/tex]:
- Combine [tex]\(-40x^2\)[/tex] and [tex]\(36x^2\)[/tex] to get [tex]\(-4x^2\)[/tex].
- The terms involving [tex]\(x\)[/tex]:
- Combine [tex]\(-30x\)[/tex] and [tex]\(48x\)[/tex] to get [tex]\(18x\)[/tex].
- The constant terms:
- The only constant is [tex]\(-40\)[/tex].
3. Write the result as a single polynomial:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
The correct answer is [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
This matches option B: [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
