Answer :
Let's multiply the two polynomials [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex] step-by-step:
1. Distribute each term of the first polynomial through the second polynomial:
- Multiply [tex]\(3x^2\)[/tex] with each term in the second polynomial [tex]\((x^2 - 3x + 2)\)[/tex]:
- [tex]\(3x^2 \cdot x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \cdot (-3x) = -9x^3\)[/tex]
- [tex]\(3x^2 \cdot 2 = 6x^2\)[/tex]
- Multiply [tex]\(-4x\)[/tex] with each term in the second polynomial [tex]\((x^2 - 3x + 2)\)[/tex]:
- [tex]\(-4x \cdot x^2 = -4x^3\)[/tex]
- [tex]\(-4x \cdot (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \cdot 2 = -8x\)[/tex]
- Multiply [tex]\(5\)[/tex] with each term in the second polynomial [tex]\((x^2 - 3x + 2)\)[/tex]:
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-3x) = -15x\)[/tex]
- [tex]\(5 \cdot 2 = 10\)[/tex]
2. Add all the results together:
- Combine all the terms obtained from the distribution:
[tex]\[
3x^4 + (-9x^3) + 6x^2 + (-4x^3) + 12x^2 + (-8x) + 5x^2 + (-15x) + 10
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-8x - 15x = -23x\)[/tex]
- The remaining terms are the [tex]\(x^4\)[/tex] term ([tex]\(3x^4\)[/tex]) and the constant term ([tex]\(10\)[/tex]).
Therefore, the resulting expression is:
[tex]\[ 3x^4 - 13x^3 + 23x^2 - 23x + 10 \][/tex]
This matches option C. So, the correct answer is:
[tex]\[ \text{C. } 3x^4 - 13x^3 + 23x^2 - 23x + 10 \][/tex]
1. Distribute each term of the first polynomial through the second polynomial:
- Multiply [tex]\(3x^2\)[/tex] with each term in the second polynomial [tex]\((x^2 - 3x + 2)\)[/tex]:
- [tex]\(3x^2 \cdot x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \cdot (-3x) = -9x^3\)[/tex]
- [tex]\(3x^2 \cdot 2 = 6x^2\)[/tex]
- Multiply [tex]\(-4x\)[/tex] with each term in the second polynomial [tex]\((x^2 - 3x + 2)\)[/tex]:
- [tex]\(-4x \cdot x^2 = -4x^3\)[/tex]
- [tex]\(-4x \cdot (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \cdot 2 = -8x\)[/tex]
- Multiply [tex]\(5\)[/tex] with each term in the second polynomial [tex]\((x^2 - 3x + 2)\)[/tex]:
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-3x) = -15x\)[/tex]
- [tex]\(5 \cdot 2 = 10\)[/tex]
2. Add all the results together:
- Combine all the terms obtained from the distribution:
[tex]\[
3x^4 + (-9x^3) + 6x^2 + (-4x^3) + 12x^2 + (-8x) + 5x^2 + (-15x) + 10
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-8x - 15x = -23x\)[/tex]
- The remaining terms are the [tex]\(x^4\)[/tex] term ([tex]\(3x^4\)[/tex]) and the constant term ([tex]\(10\)[/tex]).
Therefore, the resulting expression is:
[tex]\[ 3x^4 - 13x^3 + 23x^2 - 23x + 10 \][/tex]
This matches option C. So, the correct answer is:
[tex]\[ \text{C. } 3x^4 - 13x^3 + 23x^2 - 23x + 10 \][/tex]
