Answer :
Let's simplify the given polynomial expression step by step.
We have three expressions:
1. [tex]\( (5x^4 - 9x^3 + 7x - 1) \)[/tex]
2. [tex]\( (-8x^4 + 4x^2 - 3x + 2) \)[/tex]
3. [tex]\( \left(-4x^3 + 5x - 1\right)(2x - 7) \)[/tex]
Let's first simplify expression 3 by distributing the terms:
[tex]\[
(-4x^3 + 5x - 1)(2x - 7) = (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)
\][/tex]
Calculating each part:
- [tex]\((-4x^3)(2x) = -8x^4\)[/tex]
- [tex]\((-4x^3)(-7) = 28x^3\)[/tex]
- [tex]\((5x)(2x) = 10x^2\)[/tex]
- [tex]\((5x)(-7) = -35x\)[/tex]
- [tex]\((-1)(2x) = -2x\)[/tex]
- [tex]\((-1)(-7) = 7\)[/tex]
Now, combine them:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Combine like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
Now substitute back into the full expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Simplify by subtracting the third expression:
[tex]\[
= 5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
Combine 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]
Putting it all together, we get:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Therefore, the answer is [tex]\( \boxed{A} \)[/tex]: [tex]\( 5x^4 - 37x^3 - 6x^2 + 41x - 6 \)[/tex].
We have three expressions:
1. [tex]\( (5x^4 - 9x^3 + 7x - 1) \)[/tex]
2. [tex]\( (-8x^4 + 4x^2 - 3x + 2) \)[/tex]
3. [tex]\( \left(-4x^3 + 5x - 1\right)(2x - 7) \)[/tex]
Let's first simplify expression 3 by distributing the terms:
[tex]\[
(-4x^3 + 5x - 1)(2x - 7) = (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)
\][/tex]
Calculating each part:
- [tex]\((-4x^3)(2x) = -8x^4\)[/tex]
- [tex]\((-4x^3)(-7) = 28x^3\)[/tex]
- [tex]\((5x)(2x) = 10x^2\)[/tex]
- [tex]\((5x)(-7) = -35x\)[/tex]
- [tex]\((-1)(2x) = -2x\)[/tex]
- [tex]\((-1)(-7) = 7\)[/tex]
Now, combine them:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Combine like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
Now substitute back into the full expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Simplify by subtracting the third expression:
[tex]\[
= 5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
Combine 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]
Putting it all together, we get:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Therefore, the answer is [tex]\( \boxed{A} \)[/tex]: [tex]\( 5x^4 - 37x^3 - 6x^2 + 41x - 6 \)[/tex].
