Answer :
To simplify the given polynomial expression, let's break down the process step-by-step:
1. Write down the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - ((-4x^3 + 5x - 1)(2x - 7))
\][/tex]
2. Combine the first two polynomials:
- Combine like terms from [tex]\((5x^4 - 9x^3 + 7x - 1)\)[/tex] and [tex]\((-8x^4 + 4x^2 - 3x + 2)\)[/tex]:
- [tex]\(5x^4 - 8x^4 = -3x^4\)[/tex]
- [tex]\(-9x^3\)[/tex] (remains the same for now)
- [tex]\(4x^2\)[/tex] (remains the same)
- [tex]\(7x - 3x = 4x\)[/tex]
- [tex]\(-1 + 2 = 1\)[/tex]
- Combined expression so far: [tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1\)[/tex]
3. Expand the third expression:
- Expand [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]:
- Multiply [tex]\(-4x^3\)[/tex] by each term:
- [tex]\(-4x^3 \times 2x = -8x^4\)[/tex]
- [tex]\(-4x^3 \times -7 = 28x^3\)[/tex]
- Multiply [tex]\(5x\)[/tex] by each term:
- [tex]\(5x \times 2x = 10x^2\)[/tex]
- [tex]\(5x \times -7 = -35x\)[/tex]
- Multiply [tex]\(-1\)[/tex] by each term:
- [tex]\(-1 \times 2x = -2x\)[/tex]
- [tex]\(-1 \times -7 = 7\)[/tex]
- Result of expansion: [tex]\(-8x^4 + 28x^3 + 10x^2 - 35x + 7\)[/tex]
4. Subtract the expanded result from the combined expression:
- Subtract each corresponding term:
- [tex]\(-3x^4 - (-8x^4) = 5x^4\)[/tex]
- [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- [tex]\(4x - (-35x + 2x) = 4x + 35x - 2x = 41x\)[/tex]
- [tex]\(1 - 7 = -6\)[/tex]
5. Final simplified expression:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Thus, the simplified expression is [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
Correct answer:
[tex]\(B. \, 5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
1. Write down the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - ((-4x^3 + 5x - 1)(2x - 7))
\][/tex]
2. Combine the first two polynomials:
- Combine like terms from [tex]\((5x^4 - 9x^3 + 7x - 1)\)[/tex] and [tex]\((-8x^4 + 4x^2 - 3x + 2)\)[/tex]:
- [tex]\(5x^4 - 8x^4 = -3x^4\)[/tex]
- [tex]\(-9x^3\)[/tex] (remains the same for now)
- [tex]\(4x^2\)[/tex] (remains the same)
- [tex]\(7x - 3x = 4x\)[/tex]
- [tex]\(-1 + 2 = 1\)[/tex]
- Combined expression so far: [tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1\)[/tex]
3. Expand the third expression:
- Expand [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]:
- Multiply [tex]\(-4x^3\)[/tex] by each term:
- [tex]\(-4x^3 \times 2x = -8x^4\)[/tex]
- [tex]\(-4x^3 \times -7 = 28x^3\)[/tex]
- Multiply [tex]\(5x\)[/tex] by each term:
- [tex]\(5x \times 2x = 10x^2\)[/tex]
- [tex]\(5x \times -7 = -35x\)[/tex]
- Multiply [tex]\(-1\)[/tex] by each term:
- [tex]\(-1 \times 2x = -2x\)[/tex]
- [tex]\(-1 \times -7 = 7\)[/tex]
- Result of expansion: [tex]\(-8x^4 + 28x^3 + 10x^2 - 35x + 7\)[/tex]
4. Subtract the expanded result from the combined expression:
- Subtract each corresponding term:
- [tex]\(-3x^4 - (-8x^4) = 5x^4\)[/tex]
- [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- [tex]\(4x - (-35x + 2x) = 4x + 35x - 2x = 41x\)[/tex]
- [tex]\(1 - 7 = -6\)[/tex]
5. Final simplified expression:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Thus, the simplified expression is [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
Correct answer:
[tex]\(B. \, 5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
