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Answer :
To find the polynomial that represents the sum of the given polynomials, let's go through the addition step-by-step:
We have two polynomials:
1. [tex]\(6x^7 + 9x^3 - x + 9\)[/tex]
2. [tex]\(2x^7 + 7x^6 + 6\)[/tex]
We'll add them together, term by term, according to the power of [tex]\(x\)[/tex].
Step 1: Combine [tex]\(x^7\)[/tex] terms:
- In the first polynomial, the coefficient of [tex]\(x^7\)[/tex] is [tex]\(6\)[/tex].
- In the second polynomial, the coefficient of [tex]\(x^7\)[/tex] is [tex]\(2\)[/tex].
Add these together: [tex]\(6 + 2 = 8\)[/tex].
So, the [tex]\(x^7\)[/tex] term in the sum is [tex]\(8x^7\)[/tex].
Step 2: Combine [tex]\(x^6\)[/tex] terms:
- The first polynomial does not have an [tex]\(x^6\)[/tex] term, so we treat it as [tex]\(0\)[/tex].
- In the second polynomial, the coefficient of [tex]\(x^6\)[/tex] is [tex]\(7\)[/tex].
Add these together: [tex]\(0 + 7 = 7\)[/tex].
So, the [tex]\(x^6\)[/tex] term in the sum is [tex]\(7x^6\)[/tex].
Step 3: Combine [tex]\(x^3\)[/tex] terms:
- In the first polynomial, the coefficient of [tex]\(x^3\)[/tex] is [tex]\(9\)[/tex].
- The second polynomial does not have an [tex]\(x^3\)[/tex] term, so we treat it as [tex]\(0\)[/tex].
Add these together: [tex]\(9 + 0 = 9\)[/tex].
So, the [tex]\(x^3\)[/tex] term in the sum is [tex]\(9x^3\)[/tex].
Step 4: Combine [tex]\(x\)[/tex] (or [tex]\(x^1\)[/tex]) terms:
- In the first polynomial, the coefficient of [tex]\(x\)[/tex] is [tex]\(-1\)[/tex].
- The second polynomial does not have an [tex]\(x\)[/tex] term, so we treat it as [tex]\(0\)[/tex].
Add these together: [tex]\(-1 + 0 = -1\)[/tex].
So, the [tex]\(x\)[/tex] term in the sum is [tex]\(-x\)[/tex].
Step 5: Combine constant terms (or [tex]\(x^0\)[/tex] terms):
- In the first polynomial, the constant term is [tex]\(9\)[/tex].
- In the second polynomial, the constant term is [tex]\(6\)[/tex].
Add these together: [tex]\(9 + 6 = 15\)[/tex].
So, the constant term in the sum is [tex]\(15\)[/tex].
Final Result:
Putting it all together, the sum of the polynomials is:
[tex]\[8x^7 + 7x^6 + 9x^3 - x + 15\][/tex]
Now match this polynomial with the options given, and we find that it corresponds to option A. So the correct answer is:
A. [tex]\(8x^7 + 7x^6 + 9x^3 - x + 15\)[/tex]
We have two polynomials:
1. [tex]\(6x^7 + 9x^3 - x + 9\)[/tex]
2. [tex]\(2x^7 + 7x^6 + 6\)[/tex]
We'll add them together, term by term, according to the power of [tex]\(x\)[/tex].
Step 1: Combine [tex]\(x^7\)[/tex] terms:
- In the first polynomial, the coefficient of [tex]\(x^7\)[/tex] is [tex]\(6\)[/tex].
- In the second polynomial, the coefficient of [tex]\(x^7\)[/tex] is [tex]\(2\)[/tex].
Add these together: [tex]\(6 + 2 = 8\)[/tex].
So, the [tex]\(x^7\)[/tex] term in the sum is [tex]\(8x^7\)[/tex].
Step 2: Combine [tex]\(x^6\)[/tex] terms:
- The first polynomial does not have an [tex]\(x^6\)[/tex] term, so we treat it as [tex]\(0\)[/tex].
- In the second polynomial, the coefficient of [tex]\(x^6\)[/tex] is [tex]\(7\)[/tex].
Add these together: [tex]\(0 + 7 = 7\)[/tex].
So, the [tex]\(x^6\)[/tex] term in the sum is [tex]\(7x^6\)[/tex].
Step 3: Combine [tex]\(x^3\)[/tex] terms:
- In the first polynomial, the coefficient of [tex]\(x^3\)[/tex] is [tex]\(9\)[/tex].
- The second polynomial does not have an [tex]\(x^3\)[/tex] term, so we treat it as [tex]\(0\)[/tex].
Add these together: [tex]\(9 + 0 = 9\)[/tex].
So, the [tex]\(x^3\)[/tex] term in the sum is [tex]\(9x^3\)[/tex].
Step 4: Combine [tex]\(x\)[/tex] (or [tex]\(x^1\)[/tex]) terms:
- In the first polynomial, the coefficient of [tex]\(x\)[/tex] is [tex]\(-1\)[/tex].
- The second polynomial does not have an [tex]\(x\)[/tex] term, so we treat it as [tex]\(0\)[/tex].
Add these together: [tex]\(-1 + 0 = -1\)[/tex].
So, the [tex]\(x\)[/tex] term in the sum is [tex]\(-x\)[/tex].
Step 5: Combine constant terms (or [tex]\(x^0\)[/tex] terms):
- In the first polynomial, the constant term is [tex]\(9\)[/tex].
- In the second polynomial, the constant term is [tex]\(6\)[/tex].
Add these together: [tex]\(9 + 6 = 15\)[/tex].
So, the constant term in the sum is [tex]\(15\)[/tex].
Final Result:
Putting it all together, the sum of the polynomials is:
[tex]\[8x^7 + 7x^6 + 9x^3 - x + 15\][/tex]
Now match this polynomial with the options given, and we find that it corresponds to option A. So the correct answer is:
A. [tex]\(8x^7 + 7x^6 + 9x^3 - x + 15\)[/tex]
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