Answer :
Sure! Let's solve the problem of adding the two given polynomials step-by-step:
We have the polynomials:
1. [tex]\(-8x^6 + 10x^4 + 10x^2 - 1\)[/tex]
2. [tex]\(4x^4 - 7x^2 + 5x - 2\)[/tex]
We need to add these polynomials together. To do this, we combine like terms, which are terms with the same power of [tex]\(x\)[/tex].
### Step 1: Combine the [tex]\(x^6\)[/tex] terms
- The first polynomial has [tex]\(-8x^6\)[/tex].
- The second polynomial has no [tex]\(x^6\)[/tex] term.
- So, the combined [tex]\(x^6\)[/tex] term is [tex]\(-8x^6\)[/tex].
### Step 2: Combine the [tex]\(x^4\)[/tex] terms
- The first polynomial has [tex]\(10x^4\)[/tex].
- The second polynomial has [tex]\(4x^4\)[/tex].
- Combined, these make [tex]\(10x^4 + 4x^4 = 14x^4\)[/tex].
### Step 3: Combine the [tex]\(x^2\)[/tex] terms
- The first polynomial has [tex]\(10x^2\)[/tex].
- The second polynomial has [tex]\(-7x^2\)[/tex].
- Combined, these make [tex]\(10x^2 - 7x^2 = 3x^2\)[/tex].
### Step 4: Combine the [tex]\(x\)[/tex] terms
- The first polynomial has no [tex]\(x\)[/tex] term.
- The second polynomial has [tex]\(5x\)[/tex].
- So, the combined [tex]\(x\)[/tex] term is [tex]\(5x\)[/tex].
### Step 5: Combine the constant terms
- The first polynomial has [tex]\(-1\)[/tex].
- The second polynomial has [tex]\(-2\)[/tex].
- Combined, these make [tex]\(-1 - 2 = -3\)[/tex].
Putting it all together, the resulting polynomial after adding the two polynomials is:
[tex]\[
-8x^6 + 14x^4 + 3x^2 + 5x - 3
\][/tex]
This matches option b:
[tex]\[
\text{b. } -8x^6 + 14x^4 + 3x^2 + 5x - 3
\][/tex]
So, the correct answer is [tex]\(\text{b}\)[/tex].
We have the polynomials:
1. [tex]\(-8x^6 + 10x^4 + 10x^2 - 1\)[/tex]
2. [tex]\(4x^4 - 7x^2 + 5x - 2\)[/tex]
We need to add these polynomials together. To do this, we combine like terms, which are terms with the same power of [tex]\(x\)[/tex].
### Step 1: Combine the [tex]\(x^6\)[/tex] terms
- The first polynomial has [tex]\(-8x^6\)[/tex].
- The second polynomial has no [tex]\(x^6\)[/tex] term.
- So, the combined [tex]\(x^6\)[/tex] term is [tex]\(-8x^6\)[/tex].
### Step 2: Combine the [tex]\(x^4\)[/tex] terms
- The first polynomial has [tex]\(10x^4\)[/tex].
- The second polynomial has [tex]\(4x^4\)[/tex].
- Combined, these make [tex]\(10x^4 + 4x^4 = 14x^4\)[/tex].
### Step 3: Combine the [tex]\(x^2\)[/tex] terms
- The first polynomial has [tex]\(10x^2\)[/tex].
- The second polynomial has [tex]\(-7x^2\)[/tex].
- Combined, these make [tex]\(10x^2 - 7x^2 = 3x^2\)[/tex].
### Step 4: Combine the [tex]\(x\)[/tex] terms
- The first polynomial has no [tex]\(x\)[/tex] term.
- The second polynomial has [tex]\(5x\)[/tex].
- So, the combined [tex]\(x\)[/tex] term is [tex]\(5x\)[/tex].
### Step 5: Combine the constant terms
- The first polynomial has [tex]\(-1\)[/tex].
- The second polynomial has [tex]\(-2\)[/tex].
- Combined, these make [tex]\(-1 - 2 = -3\)[/tex].
Putting it all together, the resulting polynomial after adding the two polynomials is:
[tex]\[
-8x^6 + 14x^4 + 3x^2 + 5x - 3
\][/tex]
This matches option b:
[tex]\[
\text{b. } -8x^6 + 14x^4 + 3x^2 + 5x - 3
\][/tex]
So, the correct answer is [tex]\(\text{b}\)[/tex].
