College

Which polynomial represents the difference in the expression below?

[tex]\left(x^7+4x^6-x^2+2\right)-\left(2x^6+3x^5+x^2-6x\right)[/tex]

A. [tex]x^7+2x^6-3x^5-2x^2+6x+2[/tex]

B. [tex]x^7+2x^6+3x^5-2x^2-6x+2[/tex]

C. [tex]x^7+2x^6-3x^5+6x+2[/tex]

Answer :

To find the polynomial that represents the difference [tex]\((x^7 + 4x^6 - x^2 + 2) - (2x^6 + 3x^5 + x^2 - 6x)\)[/tex], follow these steps:

1. Write the first polynomial:
- [tex]\(x^7 + 4x^6 - x^2 + 2\)[/tex]

2. Write the second polynomial:
- [tex]\(2x^6 + 3x^5 + x^2 - 6x\)[/tex]

3. Subtract the second polynomial from the first:

- Subtract the coefficients of like terms:
- For [tex]\(x^7\)[/tex]: [tex]\(1 - 0 = 1\)[/tex]
- For [tex]\(x^6\)[/tex]: [tex]\(4 - 2 = 2\)[/tex]
- For [tex]\(x^5\)[/tex]: [tex]\(0 - 3 = -3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(-1 - 1 = -2\)[/tex]
- For [tex]\(x^1\)[/tex]: [tex]\(0 - (-6) = 6\)[/tex]
- Constant term: [tex]\(2 - 0 = 2\)[/tex]

4. Combine the results:

[tex]\(x^7 + 2x^6 - 3x^5 - 2x^2 + 6x + 2\)[/tex]

Thus, the polynomial representing the difference is:

A. [tex]\(x^7 + 2x^6 - 3x^5 - 2x^2 + 6x + 2\)[/tex]