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Answer :
To multiply the polynomials [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex], we'll use the distributive property, also known as the FOIL method for polynomials with more than two terms. Here's a step-by-step explanation:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\(3x^2\)[/tex] by each term in the second polynomial:
- [tex]\(3x^2 \times x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \times -3x = -9x^3\)[/tex]
- [tex]\(3x^2 \times 2 = 6x^2\)[/tex]
- Multiply [tex]\(-4x\)[/tex] by each term in the second polynomial:
- [tex]\(-4x \times x^2 = -4x^3\)[/tex]
- [tex]\(-4x \times -3x = 12x^2\)[/tex]
- [tex]\(-4x \times 2 = -8x\)[/tex]
- Multiply [tex]\(5\)[/tex] by each term in the second polynomial:
- [tex]\(5 \times x^2 = 5x^2\)[/tex]
- [tex]\(5 \times -3x = -15x\)[/tex]
- [tex]\(5 \times 2 = 10\)[/tex]
2. Combine all the terms obtained from multiplication:
[tex]\(3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10\)[/tex]
3. Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(3x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-8x - 15x = -23x\)[/tex]
- For the constant term: [tex]\(10\)[/tex]
4. Write the final simplified polynomial:
[tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex]
Matching the final result with the given options, the correct choice is [tex]\( \boxed{C} \)[/tex]: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\(3x^2\)[/tex] by each term in the second polynomial:
- [tex]\(3x^2 \times x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \times -3x = -9x^3\)[/tex]
- [tex]\(3x^2 \times 2 = 6x^2\)[/tex]
- Multiply [tex]\(-4x\)[/tex] by each term in the second polynomial:
- [tex]\(-4x \times x^2 = -4x^3\)[/tex]
- [tex]\(-4x \times -3x = 12x^2\)[/tex]
- [tex]\(-4x \times 2 = -8x\)[/tex]
- Multiply [tex]\(5\)[/tex] by each term in the second polynomial:
- [tex]\(5 \times x^2 = 5x^2\)[/tex]
- [tex]\(5 \times -3x = -15x\)[/tex]
- [tex]\(5 \times 2 = 10\)[/tex]
2. Combine all the terms obtained from multiplication:
[tex]\(3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10\)[/tex]
3. Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(3x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-8x - 15x = -23x\)[/tex]
- For the constant term: [tex]\(10\)[/tex]
4. Write the final simplified polynomial:
[tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex]
Matching the final result with the given options, the correct choice is [tex]\( \boxed{C} \)[/tex]: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
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