College

Select the correct answer.



Simplify the following polynomial expression:



[tex](5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-4x^3 + 5x - 1)(2x - 7)[/tex]



A. [tex]5x^4 - 37x^3 - 6x^2 + 41x - 6[/tex]

B. [tex]5x^4 - 37x^3 - 6x^2 + 41x - 8[/tex]

C. [tex]11x^4 - 21x^3 + 14x^2 + 33x - 6[/tex]

D. [tex]11x^4 - 21x^3 + 14x^2 + 33x - 8[/tex]

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:

  1. The first part is [tex]5x^4 - 9x^3 + 7x - 1[/tex].

  2. The second part is [tex]-8x^4 + 4x^2 - 3x + 2[/tex].

  3. 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:

  1. [tex]5x^4 - 8x^4 + 8x^4 = 5x^4[/tex]
  2. [tex]-9x^3 - 28x^3 = -37x^3[/tex]
  3. [tex]4x^2 - 10x^2 = -6x^2[/tex]
  4. [tex]7x - 3x + 37x = 41x[/tex]
  5. [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].

Other Questions