Answer :
Sure, let's simplify the polynomial expression step by step:
We have the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - \left((-4x^3 + 5x - 1)(2x - 7)\right)
\][/tex]
First, simplify the third part by expanding [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
We'll use the distributive property to expand:
[tex]\[
(-4x^3 + 5x - 1)(2x - 7) = (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)
\][/tex]
Carry out the multiplications:
[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 we'll substitute this back into the original expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
This simplifies to:
1. Combine the terms from the first and second part:
[tex]\[
5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2
\][/tex]
[tex]\[
= (5x^4 - 8x^4) + (-9x^3) + 4x^2 + (7x - 3x) + (2 - 1)
\][/tex]
[tex]\[
= -3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Now subtract the expanded third expression:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
- Distribute the negative:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
- Combine like terms:
[tex]\[
= (-3x^4 + 8x^4) + (-9x^3 - 28x^3) + (4x^2 - 10x^2) + (4x + 37x) + (1 - 7)
\][/tex]
[tex]\[
= 5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Therefore, the correct answer is option C: [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
We have the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - \left((-4x^3 + 5x - 1)(2x - 7)\right)
\][/tex]
First, simplify the third part by expanding [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
We'll use the distributive property to expand:
[tex]\[
(-4x^3 + 5x - 1)(2x - 7) = (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)
\][/tex]
Carry out the multiplications:
[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 we'll substitute this back into the original expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
This simplifies to:
1. Combine the terms from the first and second part:
[tex]\[
5x^4 - 9x^3 + 7x - 1 - 8x^4 + 4x^2 - 3x + 2
\][/tex]
[tex]\[
= (5x^4 - 8x^4) + (-9x^3) + 4x^2 + (7x - 3x) + (2 - 1)
\][/tex]
[tex]\[
= -3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Now subtract the expanded third expression:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
- Distribute the negative:
[tex]\[
-3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
- Combine like terms:
[tex]\[
= (-3x^4 + 8x^4) + (-9x^3 - 28x^3) + (4x^2 - 10x^2) + (4x + 37x) + (1 - 7)
\][/tex]
[tex]\[
= 5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
So, the simplified polynomial expression is:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Therefore, the correct answer is option C: [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
