College

What is the difference of the polynomials?

[tex]\left(5x^3 + 4x^2\right) - \left(6x^2 - 2x - 9\right)[/tex]

A. [tex]-x^3 + 6x^2 + 9[/tex]

B. [tex]-x^3 + 2x^2 - 9[/tex]

C. [tex]5x^3 - 2x^2 - 2x - 9[/tex]

D. [tex]5x^3 - 2x^2 + 2x + 9[/tex]

Answer :

To find the difference between the polynomials [tex]\((5x^3 + 4x^2) - (6x^2 - 2x - 9)\)[/tex] and [tex]\(-x^3 + 6x^2 + 9\)[/tex], we'll proceed step-by-step.

1. Subtract the first two polynomials:

- First Polynomial: [tex]\(5x^3 + 4x^2\)[/tex]
- Second Polynomial: [tex]\(6x^2 - 2x - 9\)[/tex]

We need to subtract the second polynomial from the first, which involves distributing the negative sign and then combining like terms.

[tex]\[
(5x^3 + 4x^2) - (6x^2 - 2x - 9) = 5x^3 + 4x^2 - 6x^2 + 2x + 9
\][/tex]

Combine the like terms:

[tex]\[
5x^3 + (4x^2 - 6x^2) + 2x + 9 = 5x^3 - 2x^2 + 2x + 9
\][/tex]

2. Subtract the third polynomial from the result of the first subtraction:

- Third Polynomial: [tex]\(-x^3 + 6x^2 + 9\)[/tex]

Subtract this polynomial from the result we got in step 1:

[tex]\[
(5x^3 - 2x^2 + 2x + 9) - (-x^3 + 6x^2 + 9)
\][/tex]

Distribute the negative sign and then combine like terms:

[tex]\[
5x^3 - 2x^2 + 2x + 9 + x^3 - 6x^2 - 9
\][/tex]

Combine the like terms:

[tex]\[
(5x^3 + x^3) + (-2x^2 - 6x^2) + 2x + (9 - 9) = 6x^3 - 8x^2 + 2x
\][/tex]

The final result from subtracting the polynomials is [tex]\(6x^3 - 8x^2 + 2x\)[/tex].

So, the difference of the original polynomials is [tex]\(6x^3 - 8x^2 + 2x\)[/tex].

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