Answer :
Let's simplify the given polynomial expression step-by-step:
Given expression:
[tex]\[
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-4x^3 + 5x - 1\right)(2x - 7)
\][/tex]
1. Distribute multiplication in the third part:
The expression [tex]\(\left(-4x^3 + 5x - 1\right)(2x - 7)\)[/tex] needs to be expanded first.
[tex]\[
(-4x^3)(2x) = -8x^4, \quad (-4x^3)(-7) = 28x^3
\][/tex]
[tex]\[
(5x)(2x) = 10x^2, \quad (5x)(-7) = -35x
\][/tex]
[tex]\[
(-1)(2x) = -2x, \quad (-1)(-7) = 7
\][/tex]
Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Simplify the middle terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
2. Subtract the expanded expression from the sum of the first two expressions:
Now substitute back into the main expression:
[tex]\[
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-8x^4 + 28x^3 + 10x^2 - 37x + 7\right)
\][/tex]
Combine all terms:
[tex]\[
5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2 - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Notice this involves subtracting each portion of the expanded product:
[tex]\[
= 5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
Simplify by combining like terms:
- [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - 8x^4 + 8x^4 = 5x^4\)[/tex]
- [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(7x - 3x + 37x = 41x\)[/tex]
- Constant terms: [tex]\(-1 + 2 - 7 = -6\)[/tex]
Thus, the simplified form of the polynomial is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
The correct answer is A. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
Given expression:
[tex]\[
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-4x^3 + 5x - 1\right)(2x - 7)
\][/tex]
1. Distribute multiplication in the third part:
The expression [tex]\(\left(-4x^3 + 5x - 1\right)(2x - 7)\)[/tex] needs to be expanded first.
[tex]\[
(-4x^3)(2x) = -8x^4, \quad (-4x^3)(-7) = 28x^3
\][/tex]
[tex]\[
(5x)(2x) = 10x^2, \quad (5x)(-7) = -35x
\][/tex]
[tex]\[
(-1)(2x) = -2x, \quad (-1)(-7) = 7
\][/tex]
Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Simplify the middle terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
2. Subtract the expanded expression from the sum of the first two expressions:
Now substitute back into the main expression:
[tex]\[
\left(5x^4 - 9x^3 + 7x - 1\right) + \left(-8x^4 + 4x^2 - 3x + 2\right) - \left(-8x^4 + 28x^3 + 10x^2 - 37x + 7\right)
\][/tex]
Combine all terms:
[tex]\[
5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2 - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Notice this involves subtracting each portion of the expanded product:
[tex]\[
= 5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
Simplify by combining like terms:
- [tex]\(x^4\)[/tex] terms: [tex]\(5x^4 - 8x^4 + 8x^4 = 5x^4\)[/tex]
- [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(7x - 3x + 37x = 41x\)[/tex]
- Constant terms: [tex]\(-1 + 2 - 7 = -6\)[/tex]
Thus, the simplified form of the polynomial is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
The correct answer is A. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
