Answer :
Let's multiply the two polynomials [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex] step by step to find the correct expression.
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]\(3x^2 \cdot x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \cdot (-3x) = -9x^3\)[/tex]
- [tex]\(3x^2 \cdot 2 = 6x^2\)[/tex]
- Multiply [tex]\(-4x\)[/tex] by each term in the second polynomial:
- [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] by each term in the second polynomial:
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-3x) = -15x\)[/tex]
- [tex]\(5 \cdot 2 = 10\)[/tex]
2. Add all the resulting terms together:
[tex]\[
3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10
\][/tex]
3. Combine like terms:
- The [tex]\(x^4\)[/tex] term: [tex]\(3x^4\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-8x - 15x = -23x\)[/tex]
- The constant term: [tex]\(10\)[/tex]
Putting it all together, the expanded polynomial is:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
The correct answer is C: [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]\(3x^2 \cdot x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \cdot (-3x) = -9x^3\)[/tex]
- [tex]\(3x^2 \cdot 2 = 6x^2\)[/tex]
- Multiply [tex]\(-4x\)[/tex] by each term in the second polynomial:
- [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] by each term in the second polynomial:
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-3x) = -15x\)[/tex]
- [tex]\(5 \cdot 2 = 10\)[/tex]
2. Add all the resulting terms together:
[tex]\[
3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10
\][/tex]
3. Combine like terms:
- The [tex]\(x^4\)[/tex] term: [tex]\(3x^4\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-8x - 15x = -23x\)[/tex]
- The constant term: [tex]\(10\)[/tex]
Putting it all together, the expanded polynomial is:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
The correct answer is C: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
