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Answer :
To multiply the polynomials [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex], we use the distributive property, also known as the FOIL method for binomials.
Let's multiply each term in the first polynomial by each term in the second polynomial:
1. First Term: [tex]\(3x^2 \cdot x^2 = 3x^4\)[/tex]
2. Outer Term: [tex]\(3x^2 \cdot (-3x) = -9x^3\)[/tex]
3. Inner Term: [tex]\(3x^2 \cdot 2 = 6x^2\)[/tex]
4. Second Term: [tex]\(-4x \cdot x^2 = -4x^3\)[/tex]
5. Outer Term: [tex]\(-4x \cdot (-3x) = 12x^2\)[/tex]
6. Inner Term: [tex]\(-4x \cdot 2 = -8x\)[/tex]
7. Third Term: [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
8. Outer Term: [tex]\(5 \cdot (-3x) = -15x\)[/tex]
9. Inner Term: [tex]\(5 \cdot 2 = 10\)[/tex]
Now, let's add all these results together:
- Combine all the terms:
- [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]
The final result is:
[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]
Thus, the correct option is:
C. [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex]
Let's multiply each term in the first polynomial by each term in the second polynomial:
1. First Term: [tex]\(3x^2 \cdot x^2 = 3x^4\)[/tex]
2. Outer Term: [tex]\(3x^2 \cdot (-3x) = -9x^3\)[/tex]
3. Inner Term: [tex]\(3x^2 \cdot 2 = 6x^2\)[/tex]
4. Second Term: [tex]\(-4x \cdot x^2 = -4x^3\)[/tex]
5. Outer Term: [tex]\(-4x \cdot (-3x) = 12x^2\)[/tex]
6. Inner Term: [tex]\(-4x \cdot 2 = -8x\)[/tex]
7. Third Term: [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
8. Outer Term: [tex]\(5 \cdot (-3x) = -15x\)[/tex]
9. Inner Term: [tex]\(5 \cdot 2 = 10\)[/tex]
Now, let's add all these results together:
- Combine all the terms:
- [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]
The final result is:
[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]
Thus, the correct option is:
C. [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex]
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