Answer :
Let's analyze the expression [tex]\(-5xy^3 + 9x^2y\)[/tex] and see which of the given options, when added as a first term, would result in a binomial of degree 4.
### Option 1: [tex]\(0\)[/tex]
When adding [tex]\(0\)[/tex] to the expression:
[tex]\[ -5xy^3 + 9x^2y + 0 \][/tex]
This doesn't affect the expression, so it remains:
[tex]\[ -5xy^3 + 9x^2y \][/tex]
Here, the term [tex]\(-5xy^3\)[/tex] has a degree of 4 (since [tex]\(1 + 3 = 4\)[/tex]), and the term [tex]\(9x^2y\)[/tex] has a degree of 3 (since [tex]\(2 + 1 = 3\)[/tex]). The highest degree is 4, but the expression is not a binomial; it retains two terms of different degrees.
### Option 2: [tex]\(5xy^3\)[/tex]
When adding [tex]\(5xy^3\)[/tex] to the expression:
[tex]\[ -5xy^3 + 9x^2y + 5xy^3 \][/tex]
Combining like terms:
[tex]\[ (-5 + 5)xy^3 + 9x^2y = 0 + 9x^2y = 9x^2y \][/tex]
Now, the expression only has one term:
[tex]\[ 9x^2y \][/tex]
The degree is 3 (since [tex]\(2 + 1 = 3\)[/tex]). This does not satisfy the requirement for a degree of 4.
### Option 3: [tex]\(9x^2y\)[/tex]
When adding [tex]\(9x^2y\)[/tex] to the expression:
[tex]\[ -5xy^3 + 9x^2y + 9x^2y \][/tex]
Combining like terms:
[tex]\[ -5xy^3 + (9 + 9)x^2y = -5xy^3 + 18x^2y \][/tex]
The terms have degrees 4 and 3, respectively. The highest degree is 4, but the expression is not a binomial; it has terms of different degrees.
### Option 4: [tex]\(8y^4\)[/tex]
When adding [tex]\(8y^4\)[/tex] to the expression:
[tex]\[ -5xy^3 + 9x^2y + 8y^4 \][/tex]
The new expression contains:
[tex]\[ 8y^4, -5xy^3, 9x^2y \][/tex]
The degree of [tex]\(8y^4\)[/tex] is 4, [tex]\( -5xy^3\)[/tex] is 4, and [tex]\(9x^2y\)[/tex] is 3. The highest degree terms are [tex]\(8y^4\)[/tex] and [tex]\(-5xy^3\)[/tex] with a degree of 4. Therefore, it meets the degree requirement and makes the expression a binomial of degree 4.
### Option 5: [tex]\(4xy^3\)[/tex]
When adding [tex]\(4xy^3\)[/tex] to the expression:
[tex]\[ -5xy^3 + 9x^2y + 4xy^3 \][/tex]
Combining like terms:
[tex]\[ (-5 + 4)xy^3 + 9x^2y = -xy^3 + 9x^2y \][/tex]
The terms in the expression now are [tex]\(-xy^3\)[/tex] and [tex]\(9x^2y\)[/tex]. The highest degree term is [tex]\(-xy^3\)[/tex] with a degree of 4. This makes the expression a binomial of degree 4.
### Conclusion
The options that, when added to the expression [tex]\(-5xy^3 + 9x^2y\)[/tex], make it a binomial with a degree of 4 are:
1. [tex]\(0\)[/tex]
2. [tex]\(8y^4\)[/tex]
3. [tex]\(4xy^3\)[/tex]
Therefore, the correct options are [tex]\(0, 8y^4,\)[/tex] and [tex]\(4xy^3\)[/tex].
### Option 1: [tex]\(0\)[/tex]
When adding [tex]\(0\)[/tex] to the expression:
[tex]\[ -5xy^3 + 9x^2y + 0 \][/tex]
This doesn't affect the expression, so it remains:
[tex]\[ -5xy^3 + 9x^2y \][/tex]
Here, the term [tex]\(-5xy^3\)[/tex] has a degree of 4 (since [tex]\(1 + 3 = 4\)[/tex]), and the term [tex]\(9x^2y\)[/tex] has a degree of 3 (since [tex]\(2 + 1 = 3\)[/tex]). The highest degree is 4, but the expression is not a binomial; it retains two terms of different degrees.
### Option 2: [tex]\(5xy^3\)[/tex]
When adding [tex]\(5xy^3\)[/tex] to the expression:
[tex]\[ -5xy^3 + 9x^2y + 5xy^3 \][/tex]
Combining like terms:
[tex]\[ (-5 + 5)xy^3 + 9x^2y = 0 + 9x^2y = 9x^2y \][/tex]
Now, the expression only has one term:
[tex]\[ 9x^2y \][/tex]
The degree is 3 (since [tex]\(2 + 1 = 3\)[/tex]). This does not satisfy the requirement for a degree of 4.
### Option 3: [tex]\(9x^2y\)[/tex]
When adding [tex]\(9x^2y\)[/tex] to the expression:
[tex]\[ -5xy^3 + 9x^2y + 9x^2y \][/tex]
Combining like terms:
[tex]\[ -5xy^3 + (9 + 9)x^2y = -5xy^3 + 18x^2y \][/tex]
The terms have degrees 4 and 3, respectively. The highest degree is 4, but the expression is not a binomial; it has terms of different degrees.
### Option 4: [tex]\(8y^4\)[/tex]
When adding [tex]\(8y^4\)[/tex] to the expression:
[tex]\[ -5xy^3 + 9x^2y + 8y^4 \][/tex]
The new expression contains:
[tex]\[ 8y^4, -5xy^3, 9x^2y \][/tex]
The degree of [tex]\(8y^4\)[/tex] is 4, [tex]\( -5xy^3\)[/tex] is 4, and [tex]\(9x^2y\)[/tex] is 3. The highest degree terms are [tex]\(8y^4\)[/tex] and [tex]\(-5xy^3\)[/tex] with a degree of 4. Therefore, it meets the degree requirement and makes the expression a binomial of degree 4.
### Option 5: [tex]\(4xy^3\)[/tex]
When adding [tex]\(4xy^3\)[/tex] to the expression:
[tex]\[ -5xy^3 + 9x^2y + 4xy^3 \][/tex]
Combining like terms:
[tex]\[ (-5 + 4)xy^3 + 9x^2y = -xy^3 + 9x^2y \][/tex]
The terms in the expression now are [tex]\(-xy^3\)[/tex] and [tex]\(9x^2y\)[/tex]. The highest degree term is [tex]\(-xy^3\)[/tex] with a degree of 4. This makes the expression a binomial of degree 4.
### Conclusion
The options that, when added to the expression [tex]\(-5xy^3 + 9x^2y\)[/tex], make it a binomial with a degree of 4 are:
1. [tex]\(0\)[/tex]
2. [tex]\(8y^4\)[/tex]
3. [tex]\(4xy^3\)[/tex]
Therefore, the correct options are [tex]\(0, 8y^4,\)[/tex] and [tex]\(4xy^3\)[/tex].
