Answer :
To simplify the given polynomial expression, follow these steps:
1. Identify each expression given in the problem:
- [tex]\((5x^4 - 9x^3 + 7x - 1)\)[/tex]
- [tex]\((-8x^4 + 4x^2 - 3x + 2)\)[/tex]
- [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]
2. First, simplify the product [tex]\( (-4x^3 + 5x - 1)(2x - 7) \)[/tex]:
- Distribute each term in [tex]\((-4x^3 + 5x - 1)\)[/tex] by [tex]\((2x - 7)\)[/tex].
- Multiply [tex]\(-4x^3\)[/tex] by each term in [tex]\((2x - 7)\)[/tex]:
- [tex]\(-4x^3 \times 2x = -8x^4\)[/tex]
- [tex]\(-4x^3 \times (-7) = 28x^3\)[/tex]
- Multiply [tex]\(5x\)[/tex] by each term in [tex]\((2x - 7)\)[/tex]:
- [tex]\(5x \times 2x = 10x^2\)[/tex]
- [tex]\(5x \times (-7) = -35x\)[/tex]
- Multiply [tex]\(-1\)[/tex] by each term in [tex]\((2x - 7)\)[/tex]:
- [tex]\(-1 \times 2x = -2x\)[/tex]
- [tex]\(-1 \times (-7) = 7\)[/tex]
3. Combine the results from the distribution:
- Resulting expression:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7\)[/tex]
- Simplify further:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex]
4. Now substitute back into the original problem:
- Add:
[tex]\((5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)\)[/tex]
to give:
[tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1\)[/tex]
- Subtract the expanded product:
[tex]\(- (-8x^4 + 28x^3 + 10x^2 - 37x + 7)\)[/tex]
5. Combine and simplify all terms:
- [tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1 \text{ } - \text{ (terms from product)}\)[/tex],
- Which simplifies to:
[tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
Hence, the simplified form of the given polynomial expression is:
5x⁴ - 37x³ - 6x² + 41x - 6
Thus, the correct answer is option C.
1. Identify each expression given in the problem:
- [tex]\((5x^4 - 9x^3 + 7x - 1)\)[/tex]
- [tex]\((-8x^4 + 4x^2 - 3x + 2)\)[/tex]
- [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]
2. First, simplify the product [tex]\( (-4x^3 + 5x - 1)(2x - 7) \)[/tex]:
- Distribute each term in [tex]\((-4x^3 + 5x - 1)\)[/tex] by [tex]\((2x - 7)\)[/tex].
- Multiply [tex]\(-4x^3\)[/tex] by each term in [tex]\((2x - 7)\)[/tex]:
- [tex]\(-4x^3 \times 2x = -8x^4\)[/tex]
- [tex]\(-4x^3 \times (-7) = 28x^3\)[/tex]
- Multiply [tex]\(5x\)[/tex] by each term in [tex]\((2x - 7)\)[/tex]:
- [tex]\(5x \times 2x = 10x^2\)[/tex]
- [tex]\(5x \times (-7) = -35x\)[/tex]
- Multiply [tex]\(-1\)[/tex] by each term in [tex]\((2x - 7)\)[/tex]:
- [tex]\(-1 \times 2x = -2x\)[/tex]
- [tex]\(-1 \times (-7) = 7\)[/tex]
3. Combine the results from the distribution:
- Resulting expression:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7\)[/tex]
- Simplify further:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex]
4. Now substitute back into the original problem:
- Add:
[tex]\((5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)\)[/tex]
to give:
[tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1\)[/tex]
- Subtract the expanded product:
[tex]\(- (-8x^4 + 28x^3 + 10x^2 - 37x + 7)\)[/tex]
5. Combine and simplify all terms:
- [tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1 \text{ } - \text{ (terms from product)}\)[/tex],
- Which simplifies to:
[tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
Hence, the simplified form of the given polynomial expression is:
5x⁴ - 37x³ - 6x² + 41x - 6
Thus, the correct answer is option C.
