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Answer :
To solve the multiplication of the polynomials [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex], we will use the distributive property. This involves multiplying each term of the first polynomial by each term of the second polynomial and then combining like terms.
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\(3x^2\)[/tex] with each term in [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 [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 [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. Combine all the 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 resulting polynomial is:
[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]
So the correct answer is D. [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] with each term in [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 [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 [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. Combine all the 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 resulting polynomial is:
[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]
So the correct answer is D. [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
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