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Answer :
To solve the problem of multiplying the polynomials [tex]\((8x^2 + 6x + 8)(6x - 5)\)[/tex], we need to use the distributive property. This involves multiplying each term in the first polynomial by each term in the second polynomial, and then combining like terms.
Let's break it down step-by-step:
1. Multiply the first term in the first polynomial by each term in the second polynomial:
- [tex]\(8x^2 \times 6x = 48x^3\)[/tex]
- [tex]\(8x^2 \times (-5) = -40x^2\)[/tex]
2. Multiply the second term in the first polynomial by each term in the second polynomial:
- [tex]\(6x \times 6x = 36x^2\)[/tex]
- [tex]\(6x \times (-5) = -30x\)[/tex]
3. Multiply the third term in the first polynomial by each term in the second polynomial:
- [tex]\(8 \times 6x = 48x\)[/tex]
- [tex]\(8 \times (-5) = -40\)[/tex]
4. Combine all the results from the above steps:
- [tex]\(48x^3\)[/tex] (from step 1)
- [tex]\(-40x^2 + 36x^2\)[/tex] (combine like [tex]\(x^2\)[/tex] terms)
- [tex]\(-30x + 48x\)[/tex] (combine like [tex]\(x\)[/tex] terms)
- [tex]\(-40\)[/tex] (constant term from step 3)
5. Simplify by combining like terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(-40x^2 + 36x^2 = -4x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-30x + 48x = 18x\)[/tex]
Putting it all together, the resulting polynomial is:
[tex]\[ 48x^3 - 4x^2 + 18x - 40 \][/tex]
Therefore, the correct answer is C. [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
Let's break it down step-by-step:
1. Multiply the first term in the first polynomial by each term in the second polynomial:
- [tex]\(8x^2 \times 6x = 48x^3\)[/tex]
- [tex]\(8x^2 \times (-5) = -40x^2\)[/tex]
2. Multiply the second term in the first polynomial by each term in the second polynomial:
- [tex]\(6x \times 6x = 36x^2\)[/tex]
- [tex]\(6x \times (-5) = -30x\)[/tex]
3. Multiply the third term in the first polynomial by each term in the second polynomial:
- [tex]\(8 \times 6x = 48x\)[/tex]
- [tex]\(8 \times (-5) = -40\)[/tex]
4. Combine all the results from the above steps:
- [tex]\(48x^3\)[/tex] (from step 1)
- [tex]\(-40x^2 + 36x^2\)[/tex] (combine like [tex]\(x^2\)[/tex] terms)
- [tex]\(-30x + 48x\)[/tex] (combine like [tex]\(x\)[/tex] terms)
- [tex]\(-40\)[/tex] (constant term from step 3)
5. Simplify by combining like terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(-40x^2 + 36x^2 = -4x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-30x + 48x = 18x\)[/tex]
Putting it all together, the resulting polynomial is:
[tex]\[ 48x^3 - 4x^2 + 18x - 40 \][/tex]
Therefore, the correct answer is C. [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
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