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Answer :
To multiply the polynomials [tex]\((8x^2 + 6x + 8)(6x - 5)\)[/tex], we'll use the distributive property, also known as the FOIL method for binomials, but expanded for larger polynomials. Here are the steps:
1. Distribute each term in the first polynomial to every term in the second polynomial:
- Multiply [tex]\(8x^2\)[/tex] by each term in the second polynomial:
- [tex]\(8x^2 \cdot 6x = 48x^3\)[/tex]
- [tex]\(8x^2 \cdot -5 = -40x^2\)[/tex]
- Multiply [tex]\(6x\)[/tex] by each term in the second polynomial:
- [tex]\(6x \cdot 6x = 36x^2\)[/tex]
- [tex]\(6x \cdot -5 = -30x\)[/tex]
- Multiply [tex]\(8\)[/tex] by each term in the second polynomial:
- [tex]\(8 \cdot 6x = 48x\)[/tex]
- [tex]\(8 \cdot -5 = -40\)[/tex]
2. Combine all these results:
- [tex]\(48x^3\)[/tex]
- [tex]\(-40x^2 + 36x^2\)[/tex] results in [tex]\(-4x^2\)[/tex]
- [tex]\(-30x + 48x\)[/tex] results in [tex]\(18x\)[/tex]
- [tex]\(-40\)[/tex]
3. Write down the resulting polynomial:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
This matches with choice B: [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
1. Distribute each term in the first polynomial to every term in the second polynomial:
- Multiply [tex]\(8x^2\)[/tex] by each term in the second polynomial:
- [tex]\(8x^2 \cdot 6x = 48x^3\)[/tex]
- [tex]\(8x^2 \cdot -5 = -40x^2\)[/tex]
- Multiply [tex]\(6x\)[/tex] by each term in the second polynomial:
- [tex]\(6x \cdot 6x = 36x^2\)[/tex]
- [tex]\(6x \cdot -5 = -30x\)[/tex]
- Multiply [tex]\(8\)[/tex] by each term in the second polynomial:
- [tex]\(8 \cdot 6x = 48x\)[/tex]
- [tex]\(8 \cdot -5 = -40\)[/tex]
2. Combine all these results:
- [tex]\(48x^3\)[/tex]
- [tex]\(-40x^2 + 36x^2\)[/tex] results in [tex]\(-4x^2\)[/tex]
- [tex]\(-30x + 48x\)[/tex] results in [tex]\(18x\)[/tex]
- [tex]\(-40\)[/tex]
3. Write down the resulting polynomial:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
This matches with choice B: [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
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