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Answer :
To find the product of the polynomials [tex]\(-3x^2 + 5\)[/tex] and [tex]\(x^3 + 3x^2 - x - 7\)[/tex], we need to multiply each term of the first polynomial by each term of the second polynomial and then combine like terms. Let's go through the steps:
1. Multiply [tex]\(-3x^2\)[/tex] by each term in the second polynomial:
- [tex]\((-3x^2) \cdot (x^3) = -3x^5\)[/tex]
- [tex]\((-3x^2) \cdot (3x^2) = -9x^4\)[/tex]
- [tex]\((-3x^2) \cdot (-x) = 3x^3\)[/tex]
- [tex]\((-3x^2) \cdot (-7) = 21x^2\)[/tex]
2. Multiply [tex]\(5\)[/tex] by each term in the second polynomial:
- [tex]\(5 \cdot (x^3) = 5x^3\)[/tex]
- [tex]\(5 \cdot (3x^2) = 15x^2\)[/tex]
- [tex]\(5 \cdot (-x) = -5x\)[/tex]
- [tex]\(5 \cdot (-7) = -35\)[/tex]
3. Combine all these terms:
[tex]\[
-3x^5 - 9x^4 + 3x^3 + 21x^2 + 5x^3 + 15x^2 - 5x - 35
\][/tex]
4. Combine like terms:
- The terms in [tex]\(x^5\)[/tex] are [tex]\(-3x^5\)[/tex].
- The terms in [tex]\(x^4\)[/tex] are [tex]\(-9x^4\)[/tex].
- The terms in [tex]\(x^3\)[/tex] are [tex]\(3x^3 + 5x^3 = 8x^3\)[/tex].
- The terms in [tex]\(x^2\)[/tex] are [tex]\(21x^2 + 15x^2 = 36x^2\)[/tex].
- The terms in [tex]\(x\)[/tex] are [tex]\(-5x\)[/tex].
- The constant terms are [tex]\(-35\)[/tex].
So, the resulting polynomial is:
[tex]\[
-3x^5 - 9x^4 + 8x^3 + 36x^2 - 5x - 35
\][/tex]
This matches option [tex]\(D\)[/tex], so the correct answer is:
D. [tex]\(-3x^5 - 9x^4 + 8x^3 + 36x^2 - 5x - 35\)[/tex]
1. Multiply [tex]\(-3x^2\)[/tex] by each term in the second polynomial:
- [tex]\((-3x^2) \cdot (x^3) = -3x^5\)[/tex]
- [tex]\((-3x^2) \cdot (3x^2) = -9x^4\)[/tex]
- [tex]\((-3x^2) \cdot (-x) = 3x^3\)[/tex]
- [tex]\((-3x^2) \cdot (-7) = 21x^2\)[/tex]
2. Multiply [tex]\(5\)[/tex] by each term in the second polynomial:
- [tex]\(5 \cdot (x^3) = 5x^3\)[/tex]
- [tex]\(5 \cdot (3x^2) = 15x^2\)[/tex]
- [tex]\(5 \cdot (-x) = -5x\)[/tex]
- [tex]\(5 \cdot (-7) = -35\)[/tex]
3. Combine all these terms:
[tex]\[
-3x^5 - 9x^4 + 3x^3 + 21x^2 + 5x^3 + 15x^2 - 5x - 35
\][/tex]
4. Combine like terms:
- The terms in [tex]\(x^5\)[/tex] are [tex]\(-3x^5\)[/tex].
- The terms in [tex]\(x^4\)[/tex] are [tex]\(-9x^4\)[/tex].
- The terms in [tex]\(x^3\)[/tex] are [tex]\(3x^3 + 5x^3 = 8x^3\)[/tex].
- The terms in [tex]\(x^2\)[/tex] are [tex]\(21x^2 + 15x^2 = 36x^2\)[/tex].
- The terms in [tex]\(x\)[/tex] are [tex]\(-5x\)[/tex].
- The constant terms are [tex]\(-35\)[/tex].
So, the resulting polynomial is:
[tex]\[
-3x^5 - 9x^4 + 8x^3 + 36x^2 - 5x - 35
\][/tex]
This matches option [tex]\(D\)[/tex], so the correct answer is:
D. [tex]\(-3x^5 - 9x^4 + 8x^3 + 36x^2 - 5x - 35\)[/tex]
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