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Answer :
To multiply the polynomials [tex]\((5x^2 + 2x + 8)\)[/tex] and [tex]\((7x - 6)\)[/tex], we can use the distributive property, applying each term in the first polynomial to every term in the second polynomial.
1. Distribute [tex]\(5x^2\)[/tex] to [tex]\(7x - 6\)[/tex]:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times -6 = -30x^2\)[/tex]
2. Distribute [tex]\(2x\)[/tex] to [tex]\(7x - 6\)[/tex]:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times -6 = -12x\)[/tex]
3. Distribute [tex]\(8\)[/tex] to [tex]\(7x - 6\)[/tex]:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times -6 = -48\)[/tex]
Now, let's combine all the terms:
- [tex]\(35x^3\)[/tex] is just by itself as there are no other [tex]\(x^3\)[/tex] terms.
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-12x + 56x = 44x\)[/tex].
- The constant term is [tex]\(-48\)[/tex].
Putting it all together, the product of the polynomials is:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
This matches option D.
1. Distribute [tex]\(5x^2\)[/tex] to [tex]\(7x - 6\)[/tex]:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times -6 = -30x^2\)[/tex]
2. Distribute [tex]\(2x\)[/tex] to [tex]\(7x - 6\)[/tex]:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times -6 = -12x\)[/tex]
3. Distribute [tex]\(8\)[/tex] to [tex]\(7x - 6\)[/tex]:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times -6 = -48\)[/tex]
Now, let's combine all the terms:
- [tex]\(35x^3\)[/tex] is just by itself as there are no other [tex]\(x^3\)[/tex] terms.
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-12x + 56x = 44x\)[/tex].
- The constant term is [tex]\(-48\)[/tex].
Putting it all together, the product of the polynomials is:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
This matches option D.
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