Answer :
To solve the problem of adding the two polynomial expressions [tex]\((3x^4 + 2 - 2x^3)\)[/tex] and [tex]\((4x^3 + 4x^4)\)[/tex], we need to combine like terms. Here’s a step-by-step guide to do this:
1. Identify the terms in each polynomial:
- The first polynomial, [tex]\(3x^4 + 2 - 2x^3\)[/tex], has the terms:
- [tex]\(3x^4\)[/tex] (a quartic term, meaning it's [tex]\(x\)[/tex] raised to the power of 4)
- [tex]\(-2x^3\)[/tex] (a cubic term, meaning it's [tex]\(x\)[/tex] raised to the power of 3)
- [tex]\(2\)[/tex] (a constant term)
- The second polynomial, [tex]\(4x^3 + 4x^4\)[/tex], has the terms:
- [tex]\(4x^3\)[/tex] (a cubic term)
- [tex]\(4x^4\)[/tex] (a quartic term)
2. Combine like terms:
- Quartic terms: Add the [tex]\(x^4\)[/tex] terms from both polynomials:
- [tex]\(3x^4 + 4x^4 = 7x^4\)[/tex]
- Cubic terms: Add the [tex]\(x^3\)[/tex] terms from both polynomials:
- [tex]\(-2x^3 + 4x^3 = 2x^3\)[/tex]
- Constant term: The only constant term is [tex]\(2\)[/tex].
3. Write the simplified expression:
- Combining all these, we have the expression:
[tex]\[
7x^4 + 2x^3 + 2
\][/tex]
From these steps, we can see that the result of this addition is [tex]\(\boxed{7x^4 + 2x^3 + 2}\)[/tex].
4. Identify the correct answer choice:
- Compare the simplified expression, [tex]\(7x^4 + 2x^3 + 2\)[/tex], with the answer choices provided:
- A. [tex]\(7x^4 + 2x^3 + 2\)[/tex]
Hence, the correct option is A.
1. Identify the terms in each polynomial:
- The first polynomial, [tex]\(3x^4 + 2 - 2x^3\)[/tex], has the terms:
- [tex]\(3x^4\)[/tex] (a quartic term, meaning it's [tex]\(x\)[/tex] raised to the power of 4)
- [tex]\(-2x^3\)[/tex] (a cubic term, meaning it's [tex]\(x\)[/tex] raised to the power of 3)
- [tex]\(2\)[/tex] (a constant term)
- The second polynomial, [tex]\(4x^3 + 4x^4\)[/tex], has the terms:
- [tex]\(4x^3\)[/tex] (a cubic term)
- [tex]\(4x^4\)[/tex] (a quartic term)
2. Combine like terms:
- Quartic terms: Add the [tex]\(x^4\)[/tex] terms from both polynomials:
- [tex]\(3x^4 + 4x^4 = 7x^4\)[/tex]
- Cubic terms: Add the [tex]\(x^3\)[/tex] terms from both polynomials:
- [tex]\(-2x^3 + 4x^3 = 2x^3\)[/tex]
- Constant term: The only constant term is [tex]\(2\)[/tex].
3. Write the simplified expression:
- Combining all these, we have the expression:
[tex]\[
7x^4 + 2x^3 + 2
\][/tex]
From these steps, we can see that the result of this addition is [tex]\(\boxed{7x^4 + 2x^3 + 2}\)[/tex].
4. Identify the correct answer choice:
- Compare the simplified expression, [tex]\(7x^4 + 2x^3 + 2\)[/tex], with the answer choices provided:
- A. [tex]\(7x^4 + 2x^3 + 2\)[/tex]
Hence, the correct option is A.
