College

Perform the indicated operation for each polynomial sum or difference.

1. [tex]\(\left(-9x^4 - 8x^3 - 10x\right) + \left(10x^4 + 8x^3 + 5\right) =\)[/tex]

2. [tex]\(\left(-3x^4 + 9x^3 + 2x^2\right) - \left(-6x^4 - 9x^3 - 8x\right) =\)[/tex]

Answer :

Sure! Let's go through each polynomial operation step by step.

### First Polynomial Operation: Addition

We want to perform the operation:
[tex]\[
(-9x^4 - 8x^3 - 10x) + (10x^4 + 8x^3 + 5)
\][/tex]

1. Combine the [tex]\(x^4\)[/tex] terms:
- [tex]\(-9x^4 + 10x^4 = 1x^4\)[/tex]

2. Combine the [tex]\(x^3\)[/tex] terms:
- [tex]\(-8x^3 + 8x^3 = 0x^3\)[/tex]

3. Combine the [tex]\(x\)[/tex] terms:
- [tex]\(-10x\)[/tex]

4. Combine constant terms:
- [tex]\(0 + 5 = 5\)[/tex]

Putting it all together, we have:
[tex]\[
1x^4 + 0x^3 + 0x^2 - 10x + 5
\][/tex]
So, the result is:
[tex]\[
x^4 - 10x + 5
\][/tex]

### Second Polynomial Operation: Subtraction

We want to perform the operation:
[tex]\[
(-3x^4 + 9x^3 + 2x^2) - (-6x^4 - 9x^3 - 8x)
\][/tex]

1. Combine the [tex]\(x^4\)[/tex] terms:
- [tex]\(-3x^4 - (-6x^4) = -3x^4 + 6x^4 = 3x^4\)[/tex]

2. Combine the [tex]\(x^3\)[/tex] terms:
- [tex]\(9x^3 - (-9x^3) = 9x^3 + 9x^3 = 18x^3\)[/tex]

3. Combine the [tex]\(x^2\)[/tex] terms:
- [tex]\(2x^2 - 0 = 2x^2\)[/tex]

4. Combine the [tex]\(x\)[/tex] terms:
- [tex]\(0 - (-8x) = 0 + 8x = 8x\)[/tex]

Putting it all together, we have:
[tex]\[
3x^4 + 18x^3 + 2x^2 + 8x
\][/tex]
So, the result is:
[tex]\[
3x^4 + 18x^3 + 2x^2 + 8x
\][/tex]

That's it! These are the results for each operation on the polynomials.

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