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Answer :
Sure! Let's go through multiplying the polynomials [tex]\(x^3 + 2x^2\)[/tex] and [tex]\(2x^3 + 3x^2 - 4x\)[/tex] step-by-step vertically.
1. Write down the polynomials:
- First polynomial: [tex]\(x^3 + 2x^2\)[/tex]
- Second polynomial: [tex]\(2x^3 + 3x^2 - 4x\)[/tex]
2. Multiply each term of the first polynomial with each term of the second polynomial:
- Multiply [tex]\(x^3\)[/tex] by each term of [tex]\(2x^3 + 3x^2 - 4x\)[/tex]:
- [tex]\(x^3 \times 2x^3 = 2x^6\)[/tex]
- [tex]\(x^3 \times 3x^2 = 3x^5\)[/tex]
- [tex]\(x^3 \times (-4x) = -4x^4\)[/tex]
- Multiply [tex]\(2x^2\)[/tex] by each term of [tex]\(2x^3 + 3x^2 - 4x\)[/tex]:
- [tex]\(2x^2 \times 2x^3 = 4x^5\)[/tex]
- [tex]\(2x^2 \times 3x^2 = 6x^4\)[/tex]
- [tex]\(2x^2 \times (-4x) = -8x^3\)[/tex]
3. Combine like terms:
- The highest degree term is [tex]\(2x^6\)[/tex].
- For [tex]\(x^5\)[/tex] terms: [tex]\(3x^5 + 4x^5 = 7x^5\)[/tex].
- For [tex]\(x^4\)[/tex] terms: [tex]\(-4x^4 + 6x^4 = 2x^4\)[/tex].
- The only [tex]\(x^3\)[/tex] term is [tex]\(-8x^3\)[/tex].
4. Write the final polynomial:
[tex]\[
2x^6 + 7x^5 + 2x^4 - 8x^3
\][/tex]
Now, based on the calculations above, the answer matches option b: [tex]\(2x^6 + 7x^5 + 2x^4 - 8x^3\)[/tex].
1. Write down the polynomials:
- First polynomial: [tex]\(x^3 + 2x^2\)[/tex]
- Second polynomial: [tex]\(2x^3 + 3x^2 - 4x\)[/tex]
2. Multiply each term of the first polynomial with each term of the second polynomial:
- Multiply [tex]\(x^3\)[/tex] by each term of [tex]\(2x^3 + 3x^2 - 4x\)[/tex]:
- [tex]\(x^3 \times 2x^3 = 2x^6\)[/tex]
- [tex]\(x^3 \times 3x^2 = 3x^5\)[/tex]
- [tex]\(x^3 \times (-4x) = -4x^4\)[/tex]
- Multiply [tex]\(2x^2\)[/tex] by each term of [tex]\(2x^3 + 3x^2 - 4x\)[/tex]:
- [tex]\(2x^2 \times 2x^3 = 4x^5\)[/tex]
- [tex]\(2x^2 \times 3x^2 = 6x^4\)[/tex]
- [tex]\(2x^2 \times (-4x) = -8x^3\)[/tex]
3. Combine like terms:
- The highest degree term is [tex]\(2x^6\)[/tex].
- For [tex]\(x^5\)[/tex] terms: [tex]\(3x^5 + 4x^5 = 7x^5\)[/tex].
- For [tex]\(x^4\)[/tex] terms: [tex]\(-4x^4 + 6x^4 = 2x^4\)[/tex].
- The only [tex]\(x^3\)[/tex] term is [tex]\(-8x^3\)[/tex].
4. Write the final polynomial:
[tex]\[
2x^6 + 7x^5 + 2x^4 - 8x^3
\][/tex]
Now, based on the calculations above, the answer matches option b: [tex]\(2x^6 + 7x^5 + 2x^4 - 8x^3\)[/tex].
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