Answer :
Let's simplify the given polynomial expression step-by-step.
The expression is:
[tex]\((5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - ((-4x^3 + 5x - 1)(2x - 7))\)[/tex].
1. Combine the first two polynomial expressions:
[tex]\((5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)\)[/tex].
- Combine like terms:
- [tex]\(5x^4 + (-8x^4) = -3x^4\)[/tex].
- [tex]\(-9x^3 + 0 = -9x^3\)[/tex].
- [tex]\(0x^2 + 4x^2 = 4x^2\)[/tex].
- [tex]\(7x + (-3x) = 4x\)[/tex].
- [tex]\(-1 + 2 = 1\)[/tex].
This results in: [tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1\)[/tex].
2. Expand and simplify the third part:
[tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
- Distribute each term:
[tex]\[
\begin{align*}
&(-4x^3) \cdot 2x = -8x^4, \\
&(-4x^3) \cdot (-7) = 28x^3, \\
&5x \cdot 2x = 10x^2, \\
&5x \cdot (-7) = -35x, \\
&(-1) \cdot 2x = -2x, \\
&(-1) \cdot (-7) = 7.
\end{align*}
\][/tex]
- Combine like terms:
- Resulting expression: [tex]\(-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7\)[/tex].
- Simplify: [tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex].
3. Subtract the expanded result from the combined first two polynomials:
[tex]\((-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)\)[/tex].
- Reverse the signs of the second expression:
[tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7\)[/tex].
- Combine like terms:
- [tex]\(-3x^4 + 8x^4 = 5x^4\)[/tex].
- [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex].
- [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex].
- [tex]\(4x + 37x = 41x\)[/tex].
- [tex]\(1 - 7 = -6\)[/tex].
The simplified expression is: [tex]\(\boxed{5x^4 - 37x^3 - 6x^2 + 41x - 6}\)[/tex].
Therefore, the correct answer is A. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
The expression is:
[tex]\((5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - ((-4x^3 + 5x - 1)(2x - 7))\)[/tex].
1. Combine the first two polynomial expressions:
[tex]\((5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2)\)[/tex].
- Combine like terms:
- [tex]\(5x^4 + (-8x^4) = -3x^4\)[/tex].
- [tex]\(-9x^3 + 0 = -9x^3\)[/tex].
- [tex]\(0x^2 + 4x^2 = 4x^2\)[/tex].
- [tex]\(7x + (-3x) = 4x\)[/tex].
- [tex]\(-1 + 2 = 1\)[/tex].
This results in: [tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1\)[/tex].
2. Expand and simplify the third part:
[tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex].
- Distribute each term:
[tex]\[
\begin{align*}
&(-4x^3) \cdot 2x = -8x^4, \\
&(-4x^3) \cdot (-7) = 28x^3, \\
&5x \cdot 2x = 10x^2, \\
&5x \cdot (-7) = -35x, \\
&(-1) \cdot 2x = -2x, \\
&(-1) \cdot (-7) = 7.
\end{align*}
\][/tex]
- Combine like terms:
- Resulting expression: [tex]\(-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7\)[/tex].
- Simplify: [tex]\(-8x^4 + 28x^3 + 10x^2 - 37x + 7\)[/tex].
3. Subtract the expanded result from the combined first two polynomials:
[tex]\((-3x^4 - 9x^3 + 4x^2 + 4x + 1) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)\)[/tex].
- Reverse the signs of the second expression:
[tex]\(-3x^4 - 9x^3 + 4x^2 + 4x + 1 + 8x^4 - 28x^3 - 10x^2 + 37x - 7\)[/tex].
- Combine like terms:
- [tex]\(-3x^4 + 8x^4 = 5x^4\)[/tex].
- [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex].
- [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex].
- [tex]\(4x + 37x = 41x\)[/tex].
- [tex]\(1 - 7 = -6\)[/tex].
The simplified expression is: [tex]\(\boxed{5x^4 - 37x^3 - 6x^2 + 41x - 6}\)[/tex].
Therefore, the correct answer is A. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].
