Answer :
To solve the multiplication of the two polynomials [tex]\((x^2 + 4x + 2)\)[/tex] and [tex]\((2x^2 + 3x - 4)\)[/tex], we proceed as follows:
1. Arrange and Write the Polynomials:
- First polynomial: [tex]\(x^2 + 4x + 2\)[/tex]
- Second polynomial: [tex]\(2x^2 + 3x - 4\)[/tex]
2. Distribute Each Term:
Multiply each term in the first polynomial by each term in the second polynomial. Approach this step-by-step:
- Multiply [tex]\(x^2\)[/tex] with each term in the second polynomial:
- [tex]\(x^2 \cdot 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \cdot 3x = 3x^3\)[/tex]
- [tex]\(x^2 \cdot (-4) = -4x^2\)[/tex]
- Multiply [tex]\(4x\)[/tex] with each term in the second polynomial:
- [tex]\(4x \cdot 2x^2 = 8x^3\)[/tex]
- [tex]\(4x \cdot 3x = 12x^2\)[/tex]
- [tex]\(4x \cdot (-4) = -16x\)[/tex]
- Multiply [tex]\(2\)[/tex] with each term in the second polynomial:
- [tex]\(2 \cdot 2x^2 = 4x^2\)[/tex]
- [tex]\(2 \cdot 3x = 6x\)[/tex]
- [tex]\(2 \cdot (-4) = -8\)[/tex]
3. Combine Like Terms:
Combine the terms obtained from the previous step by organizing them according to descending powers of [tex]\(x\)[/tex]:
- Combine all [tex]\(x^4\)[/tex] terms: [tex]\(2x^4\)[/tex]
- Combine all [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- Combine all [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- Combine all [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- Combine all constant terms: [tex]\(-8\)[/tex]
4. Write the Resulting Polynomial:
Combining all terms, we get:
[tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex]
So, the answer is B. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
1. Arrange and Write the Polynomials:
- First polynomial: [tex]\(x^2 + 4x + 2\)[/tex]
- Second polynomial: [tex]\(2x^2 + 3x - 4\)[/tex]
2. Distribute Each Term:
Multiply each term in the first polynomial by each term in the second polynomial. Approach this step-by-step:
- Multiply [tex]\(x^2\)[/tex] with each term in the second polynomial:
- [tex]\(x^2 \cdot 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \cdot 3x = 3x^3\)[/tex]
- [tex]\(x^2 \cdot (-4) = -4x^2\)[/tex]
- Multiply [tex]\(4x\)[/tex] with each term in the second polynomial:
- [tex]\(4x \cdot 2x^2 = 8x^3\)[/tex]
- [tex]\(4x \cdot 3x = 12x^2\)[/tex]
- [tex]\(4x \cdot (-4) = -16x\)[/tex]
- Multiply [tex]\(2\)[/tex] with each term in the second polynomial:
- [tex]\(2 \cdot 2x^2 = 4x^2\)[/tex]
- [tex]\(2 \cdot 3x = 6x\)[/tex]
- [tex]\(2 \cdot (-4) = -8\)[/tex]
3. Combine Like Terms:
Combine the terms obtained from the previous step by organizing them according to descending powers of [tex]\(x\)[/tex]:
- Combine all [tex]\(x^4\)[/tex] terms: [tex]\(2x^4\)[/tex]
- Combine all [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- Combine all [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- Combine all [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- Combine all constant terms: [tex]\(-8\)[/tex]
4. Write the Resulting Polynomial:
Combining all terms, we get:
[tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex]
So, the answer is B. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
