Answer :
Sure! Let's simplify the given polynomial expression step-by-step to find the correct answer.
First, let's write down the expression we need to simplify:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - ((-4x^3 + 5x - 1)(2x - 7))
\][/tex]
### Step 1: Simplify the multiplication
Let's start by simplifying the product [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
[tex]\[
(-4x^3 + 5x - 1)(2x - 7)
\][/tex]
Using the distributive property (FOIL method for binomials):
[tex]\[
= (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)
\][/tex]
We get:
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Now, combine like terms in the resulting polynomial:
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
### Step 2: Combine all the terms
Now, let's rewrite the original expression with 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]
Remove the parentheses and distribute the subtraction:
[tex]\[
5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
### Step 3: Combine the like terms
Now, let's combine the like terms:
[tex]\[
(5x^4 - 8x^4 + 8x^4) + (-9x^3 - 28x^3) + (4x^2 - 10x^2) + (7x - 3x + 37x) + (-1 + 2 - 7)
\][/tex]
Simplify each group:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
### Conclusion
The simplified form of the polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the correct answer is:
A. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
First, let's write down the expression we need to simplify:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - ((-4x^3 + 5x - 1)(2x - 7))
\][/tex]
### Step 1: Simplify the multiplication
Let's start by simplifying the product [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
[tex]\[
(-4x^3 + 5x - 1)(2x - 7)
\][/tex]
Using the distributive property (FOIL method for binomials):
[tex]\[
= (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)
\][/tex]
We get:
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Now, combine like terms in the resulting polynomial:
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
### Step 2: Combine all the terms
Now, let's rewrite the original expression with 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]
Remove the parentheses and distribute the subtraction:
[tex]\[
5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
### Step 3: Combine the like terms
Now, let's combine the like terms:
[tex]\[
(5x^4 - 8x^4 + 8x^4) + (-9x^3 - 28x^3) + (4x^2 - 10x^2) + (7x - 3x + 37x) + (-1 + 2 - 7)
\][/tex]
Simplify each group:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
### Conclusion
The simplified form of the polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the correct answer is:
A. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
