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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 need to use the distributive property, also known as the FOIL method when dealing with binomials. Here's a step-by-step breakdown:
1. Expand the expression: Multiply each term in the first polynomial by each term in the second polynomial.
- Multiply [tex]\(3x^2\)[/tex] with 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] with 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] with 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. Combine like terms: Gather all similar terms together and add them.
- Combine all [tex]\(x^4\)[/tex] terms: [tex]\(3x^4\)[/tex]
- Combine all [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- Combine all [tex]\(x^2\)[/tex] terms: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- Combine all [tex]\(x\)[/tex] terms: [tex]\(-8x - 15x = -23x\)[/tex]
- Combine constant terms: [tex]\(10\)[/tex]
Putting it all together, we have:
[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]
Hence, the correct answer to the multiplication is:
[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]
The correct choice from the provided options is:
D. [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex]
1. Expand the expression: Multiply each term in the first polynomial by each term in the second polynomial.
- Multiply [tex]\(3x^2\)[/tex] with 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] with 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] with 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. Combine like terms: Gather all similar terms together and add them.
- Combine all [tex]\(x^4\)[/tex] terms: [tex]\(3x^4\)[/tex]
- Combine all [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- Combine all [tex]\(x^2\)[/tex] terms: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- Combine all [tex]\(x\)[/tex] terms: [tex]\(-8x - 15x = -23x\)[/tex]
- Combine constant terms: [tex]\(10\)[/tex]
Putting it all together, we have:
[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]
Hence, the correct answer to the multiplication is:
[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]
The correct choice from the provided options is:
D. [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex]
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