Answer :
Sure! Let's add the polynomials step-by-step. We have two polynomials:
1. [tex]\( 7x^6 + 10x^2 - 10 \)[/tex]
2. [tex]\( 3x^6 - 6x^3 + 4 \)[/tex]
To add these polynomials, we'll combine the like terms, which are terms with the same variable raised to the same power.
### Step 1: Combine the [tex]\(x^6\)[/tex] terms
- From the first polynomial, the coefficient of [tex]\(x^6\)[/tex] is 7.
- From the second polynomial, the coefficient of [tex]\(x^6\)[/tex] is 3.
- Combine them: [tex]\(7 + 3 = 10\)[/tex].
### Step 2: Combine the [tex]\(x^3\)[/tex] terms
- The first polynomial does not have an [tex]\(x^3\)[/tex] term, so its coefficient is 0.
- From the second polynomial, the coefficient of [tex]\(x^3\)[/tex] is -6.
- Combine them: [tex]\(0 - 6 = -6\)[/tex].
### Step 3: Combine the [tex]\(x^2\)[/tex] terms
- From the first polynomial, the coefficient of [tex]\(x^2\)[/tex] is 10.
- The second polynomial does not have an [tex]\(x^2\)[/tex] term, so its coefficient is 0.
- Combine them: [tex]\(10 + 0 = 10\)[/tex].
### Step 4: Combine the constant terms
- The constant term from the first polynomial is -10.
- The constant term from the second polynomial is 4.
- Combine them: [tex]\(-10 + 4 = -6\)[/tex].
### Final Expression
Now, putting it all together, the sum of the polynomials is:
[tex]\[ 10x^6 - 6x^3 + 10x^2 - 6 \][/tex]
The correct answer matching this expression is:
[tex]\[ 10x^6 - 6x^3 + 10x^2 - 6 \][/tex]
1. [tex]\( 7x^6 + 10x^2 - 10 \)[/tex]
2. [tex]\( 3x^6 - 6x^3 + 4 \)[/tex]
To add these polynomials, we'll combine the like terms, which are terms with the same variable raised to the same power.
### Step 1: Combine the [tex]\(x^6\)[/tex] terms
- From the first polynomial, the coefficient of [tex]\(x^6\)[/tex] is 7.
- From the second polynomial, the coefficient of [tex]\(x^6\)[/tex] is 3.
- Combine them: [tex]\(7 + 3 = 10\)[/tex].
### Step 2: Combine the [tex]\(x^3\)[/tex] terms
- The first polynomial does not have an [tex]\(x^3\)[/tex] term, so its coefficient is 0.
- From the second polynomial, the coefficient of [tex]\(x^3\)[/tex] is -6.
- Combine them: [tex]\(0 - 6 = -6\)[/tex].
### Step 3: Combine the [tex]\(x^2\)[/tex] terms
- From the first polynomial, the coefficient of [tex]\(x^2\)[/tex] is 10.
- The second polynomial does not have an [tex]\(x^2\)[/tex] term, so its coefficient is 0.
- Combine them: [tex]\(10 + 0 = 10\)[/tex].
### Step 4: Combine the constant terms
- The constant term from the first polynomial is -10.
- The constant term from the second polynomial is 4.
- Combine them: [tex]\(-10 + 4 = -6\)[/tex].
### Final Expression
Now, putting it all together, the sum of the polynomials is:
[tex]\[ 10x^6 - 6x^3 + 10x^2 - 6 \][/tex]
The correct answer matching this expression is:
[tex]\[ 10x^6 - 6x^3 + 10x^2 - 6 \][/tex]
