Answer :
Let's analyze the polynomial [tex]\(3^4 k - 2j k^3 + jk^3 - 2^4 k + jk^3\)[/tex] and determine its properties after it has been fully simplified.
### Step-by-Step Simplification:
1. Combine Like Terms:
We group and simplify terms involving similar powers of [tex]\(k\)[/tex] together.
The given polynomial is:
[tex]\[ 3^4 k - 2j k^3 + jk^3 - 2^4 k + jk^3 \][/tex]
Let's first evaluate the constants [tex]\(3^4\)[/tex] and [tex]\(2^4\)[/tex]:
[tex]\[ 3^4 = 81 \quad \text{and} \quad 2^4 = 16 \][/tex]
Substituting these back into the polynomial, we get:
[tex]\[ 81k - 2jk^3 + jk^3 - 16k + jk^3 \][/tex]
2. Combine Like Terms:
- Combine the [tex]\(k^1\)[/tex] terms: [tex]\(81k - 16k\)[/tex]
- Combine the [tex]\(k^3\)[/tex] terms: [tex]\(-2jk^3 + jk^3 + jk^3\)[/tex]
Combining these terms:
[tex]\[ (81 - 16)k + (-2j + j + j)k^3 \][/tex]
Which simplifies to:
[tex]\[ 65k + 0k^3 \][/tex]
Since the [tex]\(k^3\)[/tex] terms sum to zero, we have:
[tex]\[ 65k \][/tex]
### Properties of the Simplified Polynomial:
- Number of Terms:
After simplification, we have the term [tex]\(65k\)[/tex], which counts as 1 term.
- Degree:
The degree of a polynomial is the highest power of the variable [tex]\(k\)[/tex]. Here, the highest power is [tex]\(k^1\)[/tex], hence the degree is 1.
### Conclusion:
After fully simplifying the given polynomial, it has 1 term and a degree of 1.
So, the correct statement is:
It has 1 term and a degree of 1.
### Step-by-Step Simplification:
1. Combine Like Terms:
We group and simplify terms involving similar powers of [tex]\(k\)[/tex] together.
The given polynomial is:
[tex]\[ 3^4 k - 2j k^3 + jk^3 - 2^4 k + jk^3 \][/tex]
Let's first evaluate the constants [tex]\(3^4\)[/tex] and [tex]\(2^4\)[/tex]:
[tex]\[ 3^4 = 81 \quad \text{and} \quad 2^4 = 16 \][/tex]
Substituting these back into the polynomial, we get:
[tex]\[ 81k - 2jk^3 + jk^3 - 16k + jk^3 \][/tex]
2. Combine Like Terms:
- Combine the [tex]\(k^1\)[/tex] terms: [tex]\(81k - 16k\)[/tex]
- Combine the [tex]\(k^3\)[/tex] terms: [tex]\(-2jk^3 + jk^3 + jk^3\)[/tex]
Combining these terms:
[tex]\[ (81 - 16)k + (-2j + j + j)k^3 \][/tex]
Which simplifies to:
[tex]\[ 65k + 0k^3 \][/tex]
Since the [tex]\(k^3\)[/tex] terms sum to zero, we have:
[tex]\[ 65k \][/tex]
### Properties of the Simplified Polynomial:
- Number of Terms:
After simplification, we have the term [tex]\(65k\)[/tex], which counts as 1 term.
- Degree:
The degree of a polynomial is the highest power of the variable [tex]\(k\)[/tex]. Here, the highest power is [tex]\(k^1\)[/tex], hence the degree is 1.
### Conclusion:
After fully simplifying the given polynomial, it has 1 term and a degree of 1.
So, the correct statement is:
It has 1 term and a degree of 1.
