Answer :
To simplify the given polynomial expression, let's go through it step by step.
We are given:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
1. Combine like terms from the first two polynomials:
[tex]\((5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)\)[/tex]
Combine [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - 8x^4 = -3x^4\)[/tex]
Combine [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3\)[/tex]
Combine [tex]\(x^2\)[/tex] terms: [tex]\(4x^2\)[/tex]
Combine [tex]\(x\)[/tex] terms: [tex]\(7x - 3x = 4x\)[/tex]
Combine constant terms: [tex]\(-1 + 2 = 1\)[/tex]
This gives us the intermediate expression:
[tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1\)[/tex]
2. Multiply the last expression:
Calculate [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]:
Distribute each term:
- [tex]\((-4x^3) \times (2x) = -8x^4\)[/tex]
- [tex]\((-4x^3) \times (-7) = 28x^3\)[/tex]
- [tex]\(5x \times 2x = 10x^2\)[/tex]
- [tex]\(5x \times (-7) = -35x\)[/tex]
- [tex]\((-1) \times (2x) = -2x\)[/tex]
- [tex]\((-1) \times (-7) = 7\)[/tex]
Combine terms:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7\)[/tex]
Simplified, this becomes:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex]
3. Subtract the product from the previous sum:
[tex]\((-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)\)[/tex]
Distribute the negative:
[tex]\(8x^4\)[/tex], [tex]\(-28x^3\)[/tex], [tex]\(-10x^2\)[/tex], [tex]\(37x\)[/tex], [tex]\(-7\)[/tex]
Final simplification:
Combine [tex]\(x^4\)[/tex] terms: [tex]\(-3x^4 + 8x^4 = 5x^4\)[/tex]
Combine [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
Combine [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
Combine [tex]\(x\)[/tex] terms: [tex]\(4x + 37x = 41x\)[/tex]
Combine constant terms: [tex]\(1 - 7 = -6\)[/tex]
This gives us the final simplified expression:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Thus, the correct answer is option C:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
We are given:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
1. Combine like terms from the first two polynomials:
[tex]\((5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)\)[/tex]
Combine [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - 8x^4 = -3x^4\)[/tex]
Combine [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3\)[/tex]
Combine [tex]\(x^2\)[/tex] terms: [tex]\(4x^2\)[/tex]
Combine [tex]\(x\)[/tex] terms: [tex]\(7x - 3x = 4x\)[/tex]
Combine constant terms: [tex]\(-1 + 2 = 1\)[/tex]
This gives us the intermediate expression:
[tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1\)[/tex]
2. Multiply the last expression:
Calculate [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]:
Distribute each term:
- [tex]\((-4x^3) \times (2x) = -8x^4\)[/tex]
- [tex]\((-4x^3) \times (-7) = 28x^3\)[/tex]
- [tex]\(5x \times 2x = 10x^2\)[/tex]
- [tex]\(5x \times (-7) = -35x\)[/tex]
- [tex]\((-1) \times (2x) = -2x\)[/tex]
- [tex]\((-1) \times (-7) = 7\)[/tex]
Combine terms:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7\)[/tex]
Simplified, this becomes:
[tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex]
3. Subtract the product from the previous sum:
[tex]\((-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)\)[/tex]
Distribute the negative:
[tex]\(8x^4\)[/tex], [tex]\(-28x^3\)[/tex], [tex]\(-10x^2\)[/tex], [tex]\(37x\)[/tex], [tex]\(-7\)[/tex]
Final simplification:
Combine [tex]\(x^4\)[/tex] terms: [tex]\(-3x^4 + 8x^4 = 5x^4\)[/tex]
Combine [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
Combine [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
Combine [tex]\(x\)[/tex] terms: [tex]\(4x + 37x = 41x\)[/tex]
Combine constant terms: [tex]\(1 - 7 = -6\)[/tex]
This gives us the final simplified expression:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Thus, the correct answer is option C:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
