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Answer :
To multiply the two polynomials [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex], we will expand the expression by distributing each term in the first polynomial to each term in the second polynomial.
Here’s how you do it step-by-step:
1. Multiply each term of [tex]\((3x^2 - 4x + 5)\)[/tex] by every term in [tex]\((x^2 - 3x + 2)\)[/tex]:
- First term: [tex]\(3x^2\)[/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]
- Second term: [tex]\(-4x\)[/tex]:
- [tex]\(-4x \cdot x^2 = -4x^3\)[/tex]
- [tex]\(-4x \cdot (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \cdot 2 = -8x\)[/tex]
- Third term: [tex]\(5\)[/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 results:
- Collect and write down all terms:
[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]
4. Write down the final result:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
Therefore, the correct answer is C. [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
Here’s how you do it step-by-step:
1. Multiply each term of [tex]\((3x^2 - 4x + 5)\)[/tex] by every term in [tex]\((x^2 - 3x + 2)\)[/tex]:
- First term: [tex]\(3x^2\)[/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]
- Second term: [tex]\(-4x\)[/tex]:
- [tex]\(-4x \cdot x^2 = -4x^3\)[/tex]
- [tex]\(-4x \cdot (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \cdot 2 = -8x\)[/tex]
- Third term: [tex]\(5\)[/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 results:
- Collect and write down all terms:
[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]
4. Write down the final result:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
Therefore, the correct answer is C. [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
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