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 of the polynomials [tex]\((5x^3 + 4x^2) - (6x^2 - 2x - 9)\)[/tex], we need to carefully subtract each term from the polynomials.

Let's break it down step-by-step:

1. Identify the Polynomials:
- The first polynomial is [tex]\(5x^3 + 4x^2\)[/tex].
- The second polynomial is [tex]\(6x^2 - 2x - 9\)[/tex].

2. Rewrite the Polynomial Expression:
- Distribute the negative sign across the second polynomial:
[tex]\((6x^2 - 2x - 9)\)[/tex] becomes [tex]\(-6x^2 + 2x + 9\)[/tex].

3. Set Up the Subtraction:
- Combine and align like terms:
[tex]\[
(5x^3 + 4x^2 + 0x + 0) - (0x^3 + 6x^2 - 2x - 9)
\][/tex]
- After distributing, it is:
[tex]\[
5x^3 + 4x^2 + 0x + 0 - 0x^3 - 6x^2 + 2x + 9
\][/tex]

4. Perform the Subtraction:
- Subtract the coefficients of like terms:
- For [tex]\(x^3\)[/tex] terms: [tex]\(5 - 0 = 5\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(4 - 6 = -2\)[/tex]
- For [tex]\(x^1\)[/tex] terms: [tex]\(0 + 2 = 2\)[/tex]
- For the constant terms: [tex]\(0 + 9 = 9\)[/tex]

5. Write the Resulting Polynomial:
[tex]\[
5x^3 - 2x^2 + 2x + 9
\][/tex]

Thus, the difference of the given polynomials is [tex]\(5x^3 - 2x^2 + 2x + 9\)[/tex].

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