Answer :
To solve the expression [tex]\((3x^2 - 4x + 5)(x^2 - 3x + 2)\)[/tex], let's expand it step by step:
1. Distribute Each Term in the First Polynomial:
We'll distribute each of the terms from the first polynomial [tex]\((3x^2 - 4x + 5)\)[/tex] through the second polynomial [tex]\((x^2 - 3x + 2)\)[/tex].
2. Expand Each Multiplication:
- [tex]\(3x^2 \cdot (x^2 - 3x + 2)\)[/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]
- [tex]\(-4x \cdot (x^2 - 3x + 2)\)[/tex]:
- [tex]\(-4x \cdot x^2 = -4x^3\)[/tex]
- [tex]\(-4x \cdot (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \cdot 2 = -8x\)[/tex]
- [tex]\(5 \cdot (x^2 - 3x + 2)\)[/tex]:
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-3x) = -15x\)[/tex]
- [tex]\(5 \cdot 2 = 10\)[/tex]
3. Combine All the Terms:
After expanding, we combine all the terms:
[tex]\[
3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10
\][/tex]
4. Combine Like Terms:
- The term for [tex]\(x^4\)[/tex] is [tex]\(3x^4\)[/tex].
- Combine [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex].
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex].
- Combine [tex]\(x\)[/tex] terms: [tex]\(-8x - 15x = -23x\)[/tex].
- The constant term is [tex]\(10\)[/tex].
5. Write the Final Expanded Expression:
The expanded form of the expression is:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
This matches option D: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex]. So, the correct answer is D.
1. Distribute Each Term in the First Polynomial:
We'll distribute each of the terms from the first polynomial [tex]\((3x^2 - 4x + 5)\)[/tex] through the second polynomial [tex]\((x^2 - 3x + 2)\)[/tex].
2. Expand Each Multiplication:
- [tex]\(3x^2 \cdot (x^2 - 3x + 2)\)[/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]
- [tex]\(-4x \cdot (x^2 - 3x + 2)\)[/tex]:
- [tex]\(-4x \cdot x^2 = -4x^3\)[/tex]
- [tex]\(-4x \cdot (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \cdot 2 = -8x\)[/tex]
- [tex]\(5 \cdot (x^2 - 3x + 2)\)[/tex]:
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-3x) = -15x\)[/tex]
- [tex]\(5 \cdot 2 = 10\)[/tex]
3. Combine All the Terms:
After expanding, we combine all the terms:
[tex]\[
3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10
\][/tex]
4. Combine Like Terms:
- The term for [tex]\(x^4\)[/tex] is [tex]\(3x^4\)[/tex].
- Combine [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex].
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex].
- Combine [tex]\(x\)[/tex] terms: [tex]\(-8x - 15x = -23x\)[/tex].
- The constant term is [tex]\(10\)[/tex].
5. Write the Final Expanded Expression:
The expanded form of the expression is:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
This matches option D: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex]. So, the correct answer is D.
