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Answer :
To multiply the polynomials [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex], we'll use the distributive property, often called the FOIL method (First, Outside, Inside, Last), to ensure every term in the first polynomial is multiplied by every term in the second polynomial.
Let's go through the multiplication step-by-step:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- [tex]\( (3x^2) \times (x^2) = 3x^4 \)[/tex]
- [tex]\( (3x^2) \times (-3x) = -9x^3 \)[/tex]
- [tex]\( (3x^2) \times (2) = 6x^2 \)[/tex]
- [tex]\( (-4x) \times (x^2) = -4x^3 \)[/tex]
- [tex]\( (-4x) \times (-3x) = 12x^2 \)[/tex]
- [tex]\( (-4x) \times (2) = -8x \)[/tex]
- [tex]\( (5) \times (x^2) = 5x^2 \)[/tex]
- [tex]\( (5) \times (-3x) = -15x \)[/tex]
- [tex]\( (5) \times (2) = 10 \)[/tex]
2. Combine all the 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]
Putting it all together, the expanded expression is:
[tex]\[ 3x^4 - 13x^3 + 23x^2 - 23x + 10 \][/tex]
Therefore, the correct answer is option D: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
Let's go through the multiplication step-by-step:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- [tex]\( (3x^2) \times (x^2) = 3x^4 \)[/tex]
- [tex]\( (3x^2) \times (-3x) = -9x^3 \)[/tex]
- [tex]\( (3x^2) \times (2) = 6x^2 \)[/tex]
- [tex]\( (-4x) \times (x^2) = -4x^3 \)[/tex]
- [tex]\( (-4x) \times (-3x) = 12x^2 \)[/tex]
- [tex]\( (-4x) \times (2) = -8x \)[/tex]
- [tex]\( (5) \times (x^2) = 5x^2 \)[/tex]
- [tex]\( (5) \times (-3x) = -15x \)[/tex]
- [tex]\( (5) \times (2) = 10 \)[/tex]
2. Combine all the 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]
Putting it all together, the expanded expression is:
[tex]\[ 3x^4 - 13x^3 + 23x^2 - 23x + 10 \][/tex]
Therefore, the correct answer is option D: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
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