Answer :
To simplify the given polynomial expression, we'll combine like terms step by step.
Given the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
1. Expand the third part: [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
Use the distributive property:
[tex]\[
(-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Combine like terms:
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
2. Simplify the original expression:
Add the first two polynomials:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Combine like terms:
[tex]\[
= (5x^4 - 8x^4) + (-9x^3) + 4x^2 + (7x - 3x) + (2 - 1)
\][/tex]
[tex]\[
= -3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
3. Subtract the expanded expression:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Change the signs of the second polynomial:
[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 form of the polynomial expression is:
[tex]\[ 5x^4 - 37x^3 - 6x^2 + 41x - 6 \][/tex]
Thus, the correct answer is:
D. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
Given the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)
\][/tex]
1. Expand the third part: [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
Use the distributive property:
[tex]\[
(-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Combine like terms:
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
2. Simplify the original expression:
Add the first two polynomials:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)
\][/tex]
Combine like terms:
[tex]\[
= (5x^4 - 8x^4) + (-9x^3) + 4x^2 + (7x - 3x) + (2 - 1)
\][/tex]
[tex]\[
= -3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
3. Subtract the expanded expression:
[tex]\[
(-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]
Change the signs of the second polynomial:
[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 form of the polynomial expression is:
[tex]\[ 5x^4 - 37x^3 - 6x^2 + 41x - 6 \][/tex]
Thus, the correct answer is:
D. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex]
