Answer :
Sure, let's simplify the given 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]
1. Combine the first two polynomials:
- Combine like terms from [tex]\(5x^4 - 9x^3 + 7x - 1\)[/tex] and [tex]\(-8x^4 + 4x^2 - 3x + 2\)[/tex].
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) = (5x^4 - 8x^4) + (-9x^3) + 4x^2 + (7x - 3x) + (-1 + 2)
\][/tex]
[tex]\[
= -3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Expand the multiplication part:
- Expand [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
[tex]\[
= (-4x^3) \cdot (2x) + (-4x^3) \cdot (-7) + (5x) \cdot (2x) + (5x) \cdot (-7) + (-1) \cdot (2x) + (-1) \cdot (-7)
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Subtract the expanded part from the combined polynomials:
[tex]\[
[-3x^4 - 9x^3 + 4x^2 + 4x + 1] - [-8x^4 + 28x^3 + 10x^2 - 37x + 7]
\][/tex]
- Flip the signs for all terms in the second polynomial because of the subtraction:
[tex]\[
= -3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
- Now, 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]
Thus, the simplified polynomial expression is:
[tex]\[
\boxed{5x^4 - 37x^3 - 6x^2 + 41x - 6}
\][/tex]
So, the correct answer is option C.
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]
1. Combine the first two polynomials:
- Combine like terms from [tex]\(5x^4 - 9x^3 + 7x - 1\)[/tex] and [tex]\(-8x^4 + 4x^2 - 3x + 2\)[/tex].
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) = (5x^4 - 8x^4) + (-9x^3) + 4x^2 + (7x - 3x) + (-1 + 2)
\][/tex]
[tex]\[
= -3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Expand the multiplication part:
- Expand [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
[tex]\[
= (-4x^3) \cdot (2x) + (-4x^3) \cdot (-7) + (5x) \cdot (2x) + (5x) \cdot (-7) + (-1) \cdot (2x) + (-1) \cdot (-7)
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Subtract the expanded part from the combined polynomials:
[tex]\[
[-3x^4 - 9x^3 + 4x^2 + 4x + 1] - [-8x^4 + 28x^3 + 10x^2 - 37x + 7]
\][/tex]
- Flip the signs for all terms in the second polynomial because of the subtraction:
[tex]\[
= -3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7
\][/tex]
- Now, 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]
Thus, the simplified polynomial expression is:
[tex]\[
\boxed{5x^4 - 37x^3 - 6x^2 + 41x - 6}
\][/tex]
So, the correct answer is option C.
