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Answer :
To find the product of [tex]\((3x^2 + 7)(6x^2 - 4x + 5)\)[/tex], we'll use the distributive property, often referred to as the FOIL method for binomials. Here, it involves distributing each term in the first polynomial to each term in the second polynomial.
Step-by-step solution:
1. Multiply each term in [tex]\(3x^2 + 7\)[/tex] by each term in [tex]\(6x^2 - 4x + 5\)[/tex]:
- First Term: [tex]\(3x^2\)[/tex]:
- Multiply [tex]\(3x^2\)[/tex] by [tex]\(6x^2\)[/tex]:
[tex]\[3x^2 \cdot 6x^2 = 18x^4\][/tex]
- Multiply [tex]\(3x^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[3x^2 \cdot (-4x) = -12x^3\][/tex]
- Multiply [tex]\(3x^2\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[3x^2 \cdot 5 = 15x^2\][/tex]
- Second Term: [tex]\(7\)[/tex]:
- Multiply [tex]\(7\)[/tex] by [tex]\(6x^2\)[/tex]:
[tex]\[7 \cdot 6x^2 = 42x^2\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[7 \cdot (-4x) = -28x\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[7 \cdot 5 = 35\][/tex]
2. Combine all the results:
- Collect all like terms:
- [tex]\(18x^4\)[/tex] (There is only one [tex]\(x^4\)[/tex] term)
- Combine the [tex]\(x^3\)[/tex] term: [tex]\(-12x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(15x^2 + 42x^2 = 57x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-28x\)[/tex]
- Constant term: [tex]\(35\)[/tex]
3. Final Product:
The polynomial is:
[tex]\[18x^4 - 12x^3 + 57x^2 - 28x + 35\][/tex]
Therefore, the correct answer is D. [tex]\(18x^4 - 12x^3 + 57x^2 - 28x + 35\)[/tex].
Step-by-step solution:
1. Multiply each term in [tex]\(3x^2 + 7\)[/tex] by each term in [tex]\(6x^2 - 4x + 5\)[/tex]:
- First Term: [tex]\(3x^2\)[/tex]:
- Multiply [tex]\(3x^2\)[/tex] by [tex]\(6x^2\)[/tex]:
[tex]\[3x^2 \cdot 6x^2 = 18x^4\][/tex]
- Multiply [tex]\(3x^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[3x^2 \cdot (-4x) = -12x^3\][/tex]
- Multiply [tex]\(3x^2\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[3x^2 \cdot 5 = 15x^2\][/tex]
- Second Term: [tex]\(7\)[/tex]:
- Multiply [tex]\(7\)[/tex] by [tex]\(6x^2\)[/tex]:
[tex]\[7 \cdot 6x^2 = 42x^2\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[7 \cdot (-4x) = -28x\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[7 \cdot 5 = 35\][/tex]
2. Combine all the results:
- Collect all like terms:
- [tex]\(18x^4\)[/tex] (There is only one [tex]\(x^4\)[/tex] term)
- Combine the [tex]\(x^3\)[/tex] term: [tex]\(-12x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(15x^2 + 42x^2 = 57x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-28x\)[/tex]
- Constant term: [tex]\(35\)[/tex]
3. Final Product:
The polynomial is:
[tex]\[18x^4 - 12x^3 + 57x^2 - 28x + 35\][/tex]
Therefore, the correct answer is D. [tex]\(18x^4 - 12x^3 + 57x^2 - 28x + 35\)[/tex].
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