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Answer :
To multiply the polynomials [tex]\((5x^2 + 2x + 8)(7x - 6)\)[/tex], we'll use the distributive property, also known as the FOIL method for binomials, and expand each term:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- [tex]\(5x^2 \cdot 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \cdot (-6) = -30x^2\)[/tex]
- [tex]\(2x \cdot 7x = 14x^2\)[/tex]
- [tex]\(2x \cdot (-6) = -12x\)[/tex]
- [tex]\(8 \cdot 7x = 56x\)[/tex]
- [tex]\(8 \cdot (-6) = -48\)[/tex]
2. Combine like terms:
- [tex]\(35x^3\)[/tex] (there's no other [tex]\(x^3\)[/tex] term)
- For the [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- For the [tex]\(x\)[/tex] terms: [tex]\(-12x + 56x = 44x\)[/tex]
- The constant term: [tex]\(-48\)[/tex]
3. Write the final expression by combining all the terms:
The resulting polynomial is:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
Therefore, the correct answer is option B: [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex].
1. Multiply each term in the first polynomial by each term in the second polynomial:
- [tex]\(5x^2 \cdot 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \cdot (-6) = -30x^2\)[/tex]
- [tex]\(2x \cdot 7x = 14x^2\)[/tex]
- [tex]\(2x \cdot (-6) = -12x\)[/tex]
- [tex]\(8 \cdot 7x = 56x\)[/tex]
- [tex]\(8 \cdot (-6) = -48\)[/tex]
2. Combine like terms:
- [tex]\(35x^3\)[/tex] (there's no other [tex]\(x^3\)[/tex] term)
- For the [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- For the [tex]\(x\)[/tex] terms: [tex]\(-12x + 56x = 44x\)[/tex]
- The constant term: [tex]\(-48\)[/tex]
3. Write the final expression by combining all the terms:
The resulting polynomial is:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
Therefore, the correct answer is option B: [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex].
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