College

Find the difference:

[tex]\left(21x^4 + 3x^2 - 7\right) - \left(-5x^2 + 9x + 8\right)[/tex]

A. None is correct
B. [tex]21x^4 - 2x^2 + 9x + 1[/tex]
C. [tex]16x^2 - 6x - 1[/tex]
D. [tex]21x^4 + 8x^2 - 9x - 15[/tex]

Answer :

To find the difference between the two polynomials [tex]\((21x^4 + 3x^2 - 7)\)[/tex] and [tex]\((-5x^2 + 9x + 8)\)[/tex], we follow these steps:

1. Write down the given expressions:

- First polynomial: [tex]\(21x^4 + 3x^2 - 7\)[/tex].
- Second polynomial: [tex]\(-5x^2 + 9x + 8\)[/tex].

2. Apply the subtraction operation:

We need to subtract the second polynomial from the first:

[tex]\[
(21x^4 + 3x^2 - 7) - (-5x^2 + 9x + 8)
\][/tex]

3. Distribute the negative sign:

When subtracting, distribute the minus sign to each term in the second polynomial:

[tex]\[
21x^4 + 3x^2 - 7 + 5x^2 - 9x - 8
\][/tex]

4. Combine like terms:

- There is only one [tex]\(x^4\)[/tex] term: [tex]\(21x^4\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(3x^2 + 5x^2 = 8x^2\)[/tex].
- The [tex]\(x\)[/tex] term is: [tex]\(-9x\)[/tex].
- Combine the constant terms: [tex]\(-7 - 8 = -15\)[/tex].

5. Write the final polynomial:

[tex]\[
21x^4 + 8x^2 - 9x - 15
\][/tex]

This result matches the expression given as one of the options. Therefore, the difference of the polynomials is [tex]\(21x^4 + 8x^2 - 9x - 15\)[/tex].