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Answer :
To find the expression equivalent to the product of the expressions [tex]\(3x^2-7\)[/tex] and [tex]\(5x^3-8x-6\)[/tex], we'll follow these steps:
1. Multiply the Expressions: We need to multiply each term in the first expression by each term in the second expression.
2. Distribute Terms:
- [tex]\(3x^2\)[/tex] will be multiplied by each term in [tex]\(5x^3 - 8x - 6\)[/tex]:
- [tex]\(3x^2 \times 5x^3 = 15x^5\)[/tex]
- [tex]\(3x^2 \times (-8x) = -24x^3\)[/tex]
- [tex]\(3x^2 \times (-6) = -18x^2\)[/tex]
- [tex]\(-7\)[/tex] will be multiplied by each term in [tex]\(5x^3 - 8x - 6\)[/tex]:
- [tex]\(-7 \times 5x^3 = -35x^3\)[/tex]
- [tex]\(-7 \times (-8x) = 56x\)[/tex]
- [tex]\(-7 \times (-6) = 42\)[/tex]
3. Combine Like Terms:
- The terms with [tex]\(x^5\)[/tex]: [tex]\(15x^5\)[/tex]
- The terms with [tex]\(x^3\)[/tex]: [tex]\(-24x^3 - 35x^3 = -59x^3\)[/tex]
- The terms with [tex]\(x^2\)[/tex]: [tex]\(-18x^2\)[/tex]
- The terms with [tex]\(x\)[/tex]: [tex]\(56x\)[/tex]
- The constant terms: [tex]\(42\)[/tex]
4. Write the Final Expression:
- Combine all the terms to get the final expression: [tex]\(15x^5 - 59x^3 - 18x^2 + 56x + 42\)[/tex]
Now, comparing this expression with the given options:
- A: [tex]\(8x^5-56x-6\)[/tex]
- B: [tex]\(5x^3+3x^2-8x-13\)[/tex]
- C: [tex]\(15x^5-59x^3-18x^2+56x+42\)[/tex]
- D: [tex]\(15x^5-35x^3-42x^2+56x+42\)[/tex]
The correct equivalent expression is option C: [tex]\(15x^5-59x^3-18x^2+56x+42\)[/tex].
1. Multiply the Expressions: We need to multiply each term in the first expression by each term in the second expression.
2. Distribute Terms:
- [tex]\(3x^2\)[/tex] will be multiplied by each term in [tex]\(5x^3 - 8x - 6\)[/tex]:
- [tex]\(3x^2 \times 5x^3 = 15x^5\)[/tex]
- [tex]\(3x^2 \times (-8x) = -24x^3\)[/tex]
- [tex]\(3x^2 \times (-6) = -18x^2\)[/tex]
- [tex]\(-7\)[/tex] will be multiplied by each term in [tex]\(5x^3 - 8x - 6\)[/tex]:
- [tex]\(-7 \times 5x^3 = -35x^3\)[/tex]
- [tex]\(-7 \times (-8x) = 56x\)[/tex]
- [tex]\(-7 \times (-6) = 42\)[/tex]
3. Combine Like Terms:
- The terms with [tex]\(x^5\)[/tex]: [tex]\(15x^5\)[/tex]
- The terms with [tex]\(x^3\)[/tex]: [tex]\(-24x^3 - 35x^3 = -59x^3\)[/tex]
- The terms with [tex]\(x^2\)[/tex]: [tex]\(-18x^2\)[/tex]
- The terms with [tex]\(x\)[/tex]: [tex]\(56x\)[/tex]
- The constant terms: [tex]\(42\)[/tex]
4. Write the Final Expression:
- Combine all the terms to get the final expression: [tex]\(15x^5 - 59x^3 - 18x^2 + 56x + 42\)[/tex]
Now, comparing this expression with the given options:
- A: [tex]\(8x^5-56x-6\)[/tex]
- B: [tex]\(5x^3+3x^2-8x-13\)[/tex]
- C: [tex]\(15x^5-59x^3-18x^2+56x+42\)[/tex]
- D: [tex]\(15x^5-35x^3-42x^2+56x+42\)[/tex]
The correct equivalent expression is option C: [tex]\(15x^5-59x^3-18x^2+56x+42\)[/tex].
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