Answer :
To simplify the polynomial expression given in the problem, we'll follow these steps:
1. Consider the expression:
[tex]\( (5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7) \)[/tex]
2. Distribute the subtraction into the expression:
Start by expanding the part that involves subtraction:
[tex]\(-(-4x^3 + 5x - 1)(2x - 7)\)[/tex].
3. Multiply and expand:
Distribute the terms of [tex]\((-4x^3 + 5x - 1)\)[/tex] with [tex]\((2x - 7)\)[/tex]:
[tex]\[
(-4x^3 + 5x - 1) \times (2x - 7)
\][/tex]
- Multiply [tex]\(-4x^3\)[/tex] by each term in [tex]\((2x - 7)\)[/tex]:
[tex]\[
-4x^3 \times 2x = -8x^4, \quad -4x^3 \times -7 = 28x^3
\][/tex]
- Multiply [tex]\(5x\)[/tex] by each term in [tex]\((2x - 7)\)[/tex]:
[tex]\[
5x \times 2x = 10x^2, \quad 5x \times -7 = -35x
\][/tex]
- Multiply [tex]\(-1\)[/tex] by each term in [tex]\((2x - 7)\)[/tex]:
[tex]\[
-1 \times 2x = -2x, \quad -1 \times -7 = 7
\][/tex]
Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Simplify the expression:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
4. Subtract the expanded result from the original polynomial expressions:
Now, substitute 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]
5. Combine 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:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Therefore, the simplified polynomial expression is:
5x^4 - 37x^3 - 6x^2 + 41x - 6
This corresponds to option B.
1. Consider the expression:
[tex]\( (5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7) \)[/tex]
2. Distribute the subtraction into the expression:
Start by expanding the part that involves subtraction:
[tex]\(-(-4x^3 + 5x - 1)(2x - 7)\)[/tex].
3. Multiply and expand:
Distribute the terms of [tex]\((-4x^3 + 5x - 1)\)[/tex] with [tex]\((2x - 7)\)[/tex]:
[tex]\[
(-4x^3 + 5x - 1) \times (2x - 7)
\][/tex]
- Multiply [tex]\(-4x^3\)[/tex] by each term in [tex]\((2x - 7)\)[/tex]:
[tex]\[
-4x^3 \times 2x = -8x^4, \quad -4x^3 \times -7 = 28x^3
\][/tex]
- Multiply [tex]\(5x\)[/tex] by each term in [tex]\((2x - 7)\)[/tex]:
[tex]\[
5x \times 2x = 10x^2, \quad 5x \times -7 = -35x
\][/tex]
- Multiply [tex]\(-1\)[/tex] by each term in [tex]\((2x - 7)\)[/tex]:
[tex]\[
-1 \times 2x = -2x, \quad -1 \times -7 = 7
\][/tex]
Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]
Simplify the expression:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]
4. Subtract the expanded result from the original polynomial expressions:
Now, substitute 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]
5. Combine 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:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]
Therefore, the simplified polynomial expression is:
5x^4 - 37x^3 - 6x^2 + 41x - 6
This corresponds to option B.
