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Select the correct answer.

Simplify the following polynomial expression:

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

A. [tex]11x^4 - 21x^3 + 14x^2 + 33x - 6[/tex]
B. [tex]11x^4 - 21x^3 + 14x^2 + 33x - 8[/tex]
C. [tex]5x^4 - 37x^3 - 6x^2 + 41x - 8[/tex]
D. [tex]5x^4 - 37x^3 - 6x^2 + 41x - 6[/tex]

Answer :

Let's simplify the polynomial expression step by step:

We are given the expression:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - \left(-4x^3 + 5x - 1\right)(2x - 7)
\][/tex]

Step 1: Simplify [tex]\((-4x^3 + 5x - 1)(2x - 7)\)[/tex] using the distributive property (FOIL method):

- Multiply each term in the first polynomial by each term in the second polynomial:
[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]

Combine these results:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 35x - 2x + 7
\][/tex]

Simplify by combining like terms:
[tex]\[
-8x^4 + 28x^3 + 10x^2 - 37x + 7
\][/tex]

Step 2: Substitute back into the original expression and combine the polynomials:

- Original expression with distributed terms:
[tex]\[
(5x^4 - 9x^3 + 7x - 1) + (-8x^4 + 4x^2 - 3x + 2) - (-8x^4 + 28x^3 + 10x^2 - 37x + 7)
\][/tex]

- Add or subtract coefficients of like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(5x^4 - 8x^4 + 8x^4 = 5x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(-9x^3 - 28x^3 = -37x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(4x^2 - 10x^2 = -6x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(7x + 3x + 37x = 41x\)[/tex]
- For constant terms: [tex]\(-1 - 2 - 7 = -6\)[/tex]

Putting it all together:
[tex]\[
5x^4 - 37x^3 - 6x^2 + 41x - 6
\][/tex]

Therefore, the simplified polynomial expression is [tex]\(\boxed{5x^4 - 37x^3 - 6x^2 + 41x - 6}\)[/tex].

Comparing with the options given, the correct answer is D. [tex]\(5x^4 - 37x^3 - 6x^2 + 41x - 6\)[/tex].

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