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Answer :
To multiply the polynomials [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex], we can use the distributive property (also known as the FOIL method for binomials), which involves multiplying each term of the first polynomial by each term of the second polynomial.
Here's the detailed step-by-step solution:
1. Multiply each term in the first polynomial 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]
- [tex]\(-4x \times x^2 = -4x^3\)[/tex]
- [tex]\(-4x \times -3x = 12x^2\)[/tex]
- [tex]\(-4x \times 2 = -8x\)[/tex]
- [tex]\(5 \times x^2 = 5x^2\)[/tex]
- [tex]\(5 \times -3x = -15x\)[/tex]
- [tex]\(5 \times 2 = 10\)[/tex]
2. Combine all the results:
- [tex]\(3x^4\)[/tex]
- [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- [tex]\(-8x - 15x = -23x\)[/tex]
- [tex]\(10\)[/tex]
3. Putting it all together gives us:
[tex]\[ 3x^4 - 13x^3 + 23x^2 - 23x + 10 \][/tex]
Thus, the product of the polynomials is:
3x^4 - 13x^3 + 23x^2 - 23x + 10
So, the correct answer is option B: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
Here's the detailed step-by-step solution:
1. Multiply each term in the first polynomial 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]
- [tex]\(-4x \times x^2 = -4x^3\)[/tex]
- [tex]\(-4x \times -3x = 12x^2\)[/tex]
- [tex]\(-4x \times 2 = -8x\)[/tex]
- [tex]\(5 \times x^2 = 5x^2\)[/tex]
- [tex]\(5 \times -3x = -15x\)[/tex]
- [tex]\(5 \times 2 = 10\)[/tex]
2. Combine all the results:
- [tex]\(3x^4\)[/tex]
- [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- [tex]\(-8x - 15x = -23x\)[/tex]
- [tex]\(10\)[/tex]
3. Putting it all together gives us:
[tex]\[ 3x^4 - 13x^3 + 23x^2 - 23x + 10 \][/tex]
Thus, the product of the polynomials is:
3x^4 - 13x^3 + 23x^2 - 23x + 10
So, the correct answer is option B: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
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