Answer :

Sure! Let's solve the problem of adding the two polynomials step by step:

The polynomials we have are:
1. [tex]\( 8x^4 - 14x^2 + x + 6 \)[/tex]
2. [tex]\(-12x^6 - 21x^4 - 9x^2\)[/tex]

To add these polynomials, we need to combine like terms. A like term is a term with the same variable raised to the same power.

Let's align the terms based on their exponents:

- First polynomial: [tex]\( 8x^4 - 14x^2 + 1x + 6 \)[/tex]
- Second polynomial: [tex]\(-12x^6 - 21x^4 - 9x^2\)[/tex]

Notice that these are not aligned in descending order by power, so let's rewrite them aligned for clarity, filling in missing terms with coefficients of zero:

1. [tex]\( 0x^6 + 8x^4 + 0x^3 - 14x^2 + 1x + 6 \)[/tex]
2. [tex]\(-12x^6 - 21x^4 + 0x^3 - 9x^2 + 0x + 0\)[/tex]

Now, add the coefficients of the terms with the same power:

1. [tex]\(x^6\)[/tex] term: [tex]\(0 + (-12) = -12\)[/tex]
2. [tex]\(x^4\)[/tex] term: [tex]\(8 + (-21) = -13\)[/tex]
3. [tex]\(x^3\)[/tex] term: [tex]\(0 + 0 = 0\)[/tex]
4. [tex]\(x^2\)[/tex] term: [tex]\(-14 + (-9) = -23\)[/tex]
5. [tex]\(x\)[/tex] term: [tex]\(1 + 0 = 1\)[/tex]
6. Constant term: [tex]\(6 + 0 = 6\)[/tex]

Therefore, the sum of the polynomials is:

[tex]\[
-12x^6 - 13x^4 - 23x^2 + 1x + 6
\][/tex]

This is the simplified result of adding the two given polynomials!

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Rewritten by : Barada