College

What is the sum [tex]\left(6x^3 - 5x^2 + 3x - 5\right) + \left(8x^4 + 3x^3 + 5x^2 + x + 4\right)[/tex]?

A) [tex]8x^4 - 3x^3 + 10x^2 - 2x + 9[/tex]

B) [tex]8x^4 + 9x^3 + 4x - 1[/tex]

C) [tex]9x^6 + 8x^4 + 4x^2 - 1[/tex]

D) [tex]8x^4 + 3x^3 - 10x^2 + 4x - 1[/tex]

Answer :

To find the sum of the polynomials [tex]\((6x^3 - 5x^2 + 3x - 5)\)[/tex] and [tex]\((8x^4 + 3x^3 + 5x^2 + x + 4)\)[/tex], follow these steps:

1. Identify the polynomials:
- The first polynomial is [tex]\(6x^3 - 5x^2 + 3x - 5\)[/tex].
- The second polynomial is [tex]\(8x^4 + 3x^3 + 5x^2 + x + 4\)[/tex].

2. Align the polynomials by their degrees:
- The first polynomial is a cubic polynomial because its highest degree term is [tex]\(x^3\)[/tex].
- The second polynomial is a quartic polynomial because its highest degree term is [tex]\(x^4\)[/tex].

3. Rewrite each polynomial for easier addition:
- Add missing terms with zero coefficients in the first polynomial to match the second one in terms of structure:
- [tex]\(0x^4 + 6x^3 - 5x^2 + 3x - 5\)[/tex]

4. Add the corresponding coefficients:
- [tex]\(8x^4 + 0x^4 = 8x^4\)[/tex]
- [tex]\(3x^3 + 6x^3 = 9x^3\)[/tex]
- [tex]\(5x^2 - 5x^2 = 0x^2\)[/tex] (note that they cancel each other out)
- [tex]\(x + 3x = 4x\)[/tex]
- [tex]\(4 - 5 = -1\)[/tex]

5. Write the resulting polynomial:
- [tex]\(8x^4 + 9x^3 + 0x^2 + 4x - 1\)[/tex]

In simplifying, you can omit the zero term, resulting in:
[tex]\[ 8x^4 + 9x^3 + 4x - 1 \][/tex]

So, the correct answer is:
B) [tex]\(8x^4 + 9x^3 + 4x - 1\)[/tex].

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