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Answer :
To multiply the polynomials [tex]\((5x^2 + 2x + 8)(7x - 6)\)[/tex], we need to use the distributive property, also known as the FOIL method (First, Outside, Inside, Last), when dealing with binomials, but since we have a trinomial and a binomial, we'll apply the distributive property to each term in the trinomial.
Let's multiply each term in the first polynomial by each term in the second polynomial:
1. Multiply the first term of the first polynomial by both terms of the second polynomial:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times (-6) = -30x^2\)[/tex]
2. Multiply the second term of the first polynomial by both terms of the second polynomial:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times (-6) = -12x\)[/tex]
3. Multiply the third term of the first polynomial by both terms of the second polynomial:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times (-6) = -48\)[/tex]
Now, combine the like terms from the results:
- The [tex]\(x^3\)[/tex] term: [tex]\(35x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-12x + 56x = 44x\)[/tex]
- The constant term: [tex]\(-48\)[/tex]
Putting it all together, the resulting polynomial is:
[tex]\[35x^3 - 16x^2 + 44x - 48\][/tex]
So, the correct answer is option C: [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex].
Let's multiply each term in the first polynomial by each term in the second polynomial:
1. Multiply the first term of the first polynomial by both terms of the second polynomial:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times (-6) = -30x^2\)[/tex]
2. Multiply the second term of the first polynomial by both terms of the second polynomial:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times (-6) = -12x\)[/tex]
3. Multiply the third term of the first polynomial by both terms of the second polynomial:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times (-6) = -48\)[/tex]
Now, combine the like terms from the results:
- The [tex]\(x^3\)[/tex] term: [tex]\(35x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-12x + 56x = 44x\)[/tex]
- The constant term: [tex]\(-48\)[/tex]
Putting it all together, the resulting polynomial is:
[tex]\[35x^3 - 16x^2 + 44x - 48\][/tex]
So, the correct answer is option C: [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex].
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