Answer :
Sure, let's simplify the polynomial expression step-by-step.
We are given the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - \left[(-4x^3 + 5x - 1)(2x - 7)\right]
\][/tex]
### Step 1: Distribute the terms in the third part
First, let's expand the product [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]:
- Multiply [tex]\(-4x^3\)[/tex] by each term in [tex]\( (2x - 7) \)[/tex]:
[tex]\[
-4x^3 \cdot 2x = -8x^4
\][/tex]
[tex]\[
-4x^3 \cdot (-7) = 28x^3
\][/tex]
- Multiply [tex]\(5x\)[/tex] by each term in [tex]\( (2x - 7) \)[/tex]:
[tex]\[
5x \cdot 2x = 10x^2
\][/tex]
[tex]\[
5x \cdot (-7) = -35x
\][/tex]
- Multiply [tex]\(-1\)[/tex] by each term in [tex]\( (2x - 7) \)[/tex]:
[tex]\[
-1 \cdot 2x = -2x
\][/tex]
[tex]\[
-1 \cdot (-7) = 7
\][/tex]
Putting it all together, we get:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Combine like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
### Step 2: Substitute back into the expression and simplify
Substitute the expanded expression back into the original formula:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Remove the parentheses and combine like terms:
- Combine [tex]\(x^4\)[/tex] terms:
[tex]\[
5x^4 - 8x^4 + 8x^4 = 5x^4
\][/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]\[
7x - 3x + 37x = 41x
\][/tex]
- Combine constant terms:
[tex]\[
-1 + 2 - 7 = -6
\][/tex]
### Final Answer
The simplified expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the correct answer is option B:
5x^4 - 37x^3 - 6x^2 + 41x - 6.
We are given the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - \left[(-4x^3 + 5x - 1)(2x - 7)\right]
\][/tex]
### Step 1: Distribute the terms in the third part
First, let's expand the product [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex]:
- Multiply [tex]\(-4x^3\)[/tex] by each term in [tex]\( (2x - 7) \)[/tex]:
[tex]\[
-4x^3 \cdot 2x = -8x^4
\][/tex]
[tex]\[
-4x^3 \cdot (-7) = 28x^3
\][/tex]
- Multiply [tex]\(5x\)[/tex] by each term in [tex]\( (2x - 7) \)[/tex]:
[tex]\[
5x \cdot 2x = 10x^2
\][/tex]
[tex]\[
5x \cdot (-7) = -35x
\][/tex]
- Multiply [tex]\(-1\)[/tex] by each term in [tex]\( (2x - 7) \)[/tex]:
[tex]\[
-1 \cdot 2x = -2x
\][/tex]
[tex]\[
-1 \cdot (-7) = 7
\][/tex]
Putting it all together, we get:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Combine like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
### Step 2: Substitute back into the expression and simplify
Substitute the expanded expression back into the original formula:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Remove the parentheses and combine like terms:
- Combine [tex]\(x^4\)[/tex] terms:
[tex]\[
5x^4 - 8x^4 + 8x^4 = 5x^4
\][/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]\[
7x - 3x + 37x = 41x
\][/tex]
- Combine constant terms:
[tex]\[
-1 + 2 - 7 = -6
\][/tex]
### Final Answer
The simplified expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the correct answer is option B:
5x^4 - 37x^3 - 6x^2 + 41x - 6.
