Answer :
To simplify the given polynomial expression, we need to handle each term step-by-step and combine them accordingly. Here is the detailed process:
The polynomial expression given is:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - \left( (-4x^3 + 5x - 1)(2x - 7) \right)
\][/tex]
1. First, simplify directly inside the parentheses:
Combine [tex]\( (5x^4 - 9x^3 + 7x - 1) \)[/tex] and [tex]\( (-8x^4 + 4x^2 - 3x + 2) \)[/tex]:
[tex]\[
(5x^4 - 8x^4) + (-9x^3) + 4x^2 + (7x - 3x) + (-1 + 2)
\][/tex]
[tex]\[
= -3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Next, we simplify [tex]\( (-4x^3 + 5x - 1)(2x - 7) \)[/tex].
Use the distributive property (FOIL method):
[tex]\[
(-4x^3 + 5x - 1)(2x) + (-4x^3 + 5x - 1)(-7)
\][/tex]
First, distribute [tex]\( 2x \)[/tex]:
[tex]\[
-4x^3 \cdot 2x + 5x \cdot 2x - 1 \cdot 2x = -8x^4 + 10x^2 - 2x
\][/tex]
Next, distribute [tex]\( -7 \)[/tex]:
[tex]\[
-4x^3 \cdot (-7) + 5x \cdot (-7) - 1 \cdot (-7) = 28x^3 - 35x + 7
\][/tex]
Now, combine both distributions:
[tex]\[
(-8x^4) + (28x^3) + (10x^2) - (2x) - (35x) + (7)
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Now subtract this result from [tex]\( -3x^4 - 9x^3 + 4x^2 + 4x + 1 \)[/tex]:
[tex]\[
\left( -3x^4 - 9x^3 + 4x^2 + 4x + 1 \right) - \left( -8x^4 + 28x^3 + 10x^2 - 37x + 7 \right)
\][/tex]
Distribute the negative sign:
[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]
Combine each term:
[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:
[tex]\[
\boxed{B. \, 5x^4 - 37x^3 - 6x^2 + 41x - 6}
\][/tex]
The polynomial expression given is:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - \left( (-4x^3 + 5x - 1)(2x - 7) \right)
\][/tex]
1. First, simplify directly inside the parentheses:
Combine [tex]\( (5x^4 - 9x^3 + 7x - 1) \)[/tex] and [tex]\( (-8x^4 + 4x^2 - 3x + 2) \)[/tex]:
[tex]\[
(5x^4 - 8x^4) + (-9x^3) + 4x^2 + (7x - 3x) + (-1 + 2)
\][/tex]
[tex]\[
= -3x^4 - 9x^3 + 4x^2 + 4x + 1
\][/tex]
2. Next, we simplify [tex]\( (-4x^3 + 5x - 1)(2x - 7) \)[/tex].
Use the distributive property (FOIL method):
[tex]\[
(-4x^3 + 5x - 1)(2x) + (-4x^3 + 5x - 1)(-7)
\][/tex]
First, distribute [tex]\( 2x \)[/tex]:
[tex]\[
-4x^3 \cdot 2x + 5x \cdot 2x - 1 \cdot 2x = -8x^4 + 10x^2 - 2x
\][/tex]
Next, distribute [tex]\( -7 \)[/tex]:
[tex]\[
-4x^3 \cdot (-7) + 5x \cdot (-7) - 1 \cdot (-7) = 28x^3 - 35x + 7
\][/tex]
Now, combine both distributions:
[tex]\[
(-8x^4) + (28x^3) + (10x^2) - (2x) - (35x) + (7)
\][/tex]
[tex]\[
= -8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
3. Now subtract this result from [tex]\( -3x^4 - 9x^3 + 4x^2 + 4x + 1 \)[/tex]:
[tex]\[
\left( -3x^4 - 9x^3 + 4x^2 + 4x + 1 \right) - \left( -8x^4 + 28x^3 + 10x^2 - 37x + 7 \right)
\][/tex]
Distribute the negative sign:
[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]
Combine each term:
[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:
[tex]\[
\boxed{B. \, 5x^4 - 37x^3 - 6x^2 + 41x - 6}
\][/tex]
