Answer :
To simplify the polynomial expression:
[tex](5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)[/tex]
we will break it down into steps.
Step 1: Simplify each part separately:
The first part is [tex]5x^4 - 9x^3 + 7x - 1[/tex].
The second part is [tex]-8x^4 + 4x^2 - 3x + 2[/tex].
The third part involves the product [tex](-4x^3 + 5x - 1)(2x - 7)[/tex]. We need to expand this expression:
[tex](-4x^3 + 5x - 1)(2x - 7) = (-4x^3)(2x) + (-4x^3)(-7) + (5x)(2x) + (5x)(-7) + (-1)(2x) + (-1)(-7)[/tex]
Solving these terms:
- [tex](-4x^3)(2x) = -8x^4[/tex]
- [tex](-4x^3)(-7) = 28x^3[/tex]
- [tex](5x)(2x) = 10x^2[/tex]
- [tex](5x)(-7) = -35x[/tex]
- [tex](-1)(2x) = -2x[/tex]
- [tex](-1)(-7) = 7[/tex]
Combining these, we get:
[tex]-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7 = -8x^4 + 28x^3 + 10x^2 - 37x + 7[/tex]
Step 2: Substitute and simplify:
Combine all parts by substituting the expanded form 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]
Combine like terms:
- [tex]5x^4 - 8x^4 + 8x^4 = 5x^4[/tex]
- [tex]-9x^3 - 28x^3 = -37x^3[/tex]
- [tex]4x^2 - 10x^2 = -6x^2[/tex]
- [tex]7x - 3x + 37x = 41x[/tex]
- [tex]-1 - 2 - 7 = -8[/tex]
Putting it all together, we have:
[tex]5x^4 - 37x^3 - 6x^2 + 41x - 8[/tex]
Final Answer: Option B, [tex]5x^4 - 37x^3 - 6x^2 + 41x - 8[/tex].