Answer :
To multiply the polynomials [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex], we can use the distributive property, which involves distributing each term in the first polynomial across all the terms in the second polynomial. Here’s a step-by-step breakdown:
1. Multiply each term in [tex]\(x^2 - 5x\)[/tex] by each term in [tex]\(2x^2 + x - 3\)[/tex]:
- First, multiply [tex]\(x^2\)[/tex] by each term in the second polynomial:
- [tex]\(x^2 \cdot 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \cdot x = x^3\)[/tex]
- [tex]\(x^2 \cdot (-3) = -3x^2\)[/tex]
- Next, multiply [tex]\(-5x\)[/tex] by each term in the second polynomial:
- [tex]\(-5x \cdot 2x^2 = -10x^3\)[/tex]
- [tex]\(-5x \cdot x = -5x^2\)[/tex]
- [tex]\(-5x \cdot (-3) = 15x\)[/tex]
2. Combine all these results:
[tex]\(2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x\)[/tex]
3. Now, combine like terms:
- The [tex]\(x^4\)[/tex] term: [tex]\(2x^4\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
- The [tex]\(x\)[/tex] term: [tex]\(15x\)[/tex]
4. Write the resulting polynomial:
[tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
So, the correct answer is option D: [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
1. Multiply each term in [tex]\(x^2 - 5x\)[/tex] by each term in [tex]\(2x^2 + x - 3\)[/tex]:
- First, multiply [tex]\(x^2\)[/tex] by each term in the second polynomial:
- [tex]\(x^2 \cdot 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \cdot x = x^3\)[/tex]
- [tex]\(x^2 \cdot (-3) = -3x^2\)[/tex]
- Next, multiply [tex]\(-5x\)[/tex] by each term in the second polynomial:
- [tex]\(-5x \cdot 2x^2 = -10x^3\)[/tex]
- [tex]\(-5x \cdot x = -5x^2\)[/tex]
- [tex]\(-5x \cdot (-3) = 15x\)[/tex]
2. Combine all these results:
[tex]\(2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x\)[/tex]
3. Now, combine like terms:
- The [tex]\(x^4\)[/tex] term: [tex]\(2x^4\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
- The [tex]\(x\)[/tex] term: [tex]\(15x\)[/tex]
4. Write the resulting polynomial:
[tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex]
So, the correct answer is option D: [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
