High School

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'll follow these steps:

1. Write down the polynomials:
- First polynomial (inside the first set of parentheses): [tex]\(5x^3 + 4x^2\)[/tex]
- Second polynomial (inside the second set of parentheses): [tex]\(6x^2 - 2x - 9\)[/tex]

2. Distribute the negative sign (-) across the second polynomial:
- This means changing the sign of each term in the polynomial. So, [tex]\( (6x^2 - 2x - 9) \)[/tex] becomes [tex]\(-6x^2 + 2x + 9\)[/tex].

3. Subtract the polynomials by combining like terms:
- Combine the terms with the same degree from both polynomials:

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

- Group like terms:
- The [tex]\(x^3\)[/tex] terms: [tex]\(5x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(4x^2 - 6x^2 = -2x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(2x\)[/tex]
- Constant terms: [tex]\(9\)[/tex]

4. Write the final result:
- After combining the like terms, we get:
[tex]\[
5x^3 - 2x^2 + 2x + 9
\][/tex]

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