College

Which polynomial represents the sum below?

[tex]\left(8x^2 + 5x + 3\right) + \left(-5x^6 - 2x^5 - 4x^2 - 2x\right)[/tex]

A. [tex]-5x^6 - 2x^5 + 4x^2 - 8x + 2[/tex]

B. [tex]4x^6 - 2x^5 + 4x^2 + 3x + 3[/tex]

C. [tex]-5x^6 - 2x^5 + 4x^2 + 3x + 3[/tex]

D. [tex]-5x^6 - 2x^5 + 4x^2 - 3x + 3[/tex]

Answer :

To find the sum of the two given polynomials, let's break it down step by step:

We're adding these two polynomials:
1. [tex]\( (8x^2 + 5x + 3) \)[/tex]
2. [tex]\( (-5x^6 - 2x^5 - 4x^2 - 2x) \)[/tex]

Step 1: Identify and Combine Like Terms

1. [tex]\(x^6\)[/tex] terms:
- There is only one [tex]\(x^6\)[/tex] term: [tex]\(-5x^6\)[/tex].

2. [tex]\(x^5\)[/tex] terms:
- There is only one [tex]\(x^5\)[/tex] term: [tex]\(-2x^5\)[/tex].

3. [tex]\(x^2\)[/tex] terms:
- From the first polynomial, we have [tex]\(8x^2\)[/tex].
- From the second polynomial, we have [tex]\(-4x^2\)[/tex].
- Combine them: [tex]\(8x^2 + (-4x^2) = 4x^2\)[/tex].

4. [tex]\(x\)[/tex] terms:
- From the first polynomial, we have [tex]\(5x\)[/tex].
- From the second polynomial, we have [tex]\(-2x\)[/tex].
- Combine them: [tex]\(5x + (-2x) = 3x\)[/tex].

5. Constant terms:
- The constant term from the first polynomial is [tex]\(3\)[/tex].
- There is no constant term in the second polynomial, so the constant stays as [tex]\(3\)[/tex].

Step 2: Write the Resulting Polynomial

After combining the like terms, we get the polynomial:
[tex]\[ -5x^6 - 2x^5 + 4x^2 + 3x + 3 \][/tex]

This matches option C:
[tex]\[ -5x^6 - 2x^5 + 4x^2 + 3x + 3 \][/tex]

Therefore, the polynomial that represents the sum is:
C. [tex]\( -5x^6 - 2x^5 + 4x^2 + 3x + 3 \)[/tex]