To determine the missing exponent on the [tex]\( x \)[/tex]-term so that the expression [tex]\( 5x^2 y^3 + x y^2 + 8 \)[/tex] is a trinomial with a degree of 5, we need to examine each term and understand the overall degree constraints.
Step-by-Step Solution:
1. Understanding Polynomial Degrees:
- The degree of a polynomial is determined by the highest degree of its terms.
- The degree of a term is the sum of the exponents of all variables in that term.
2. Analyzing each term:
- For the term [tex]\( 5x^2 y^3 \)[/tex]:
- The exponent of [tex]\( x \)[/tex] is 2.
- The exponent of [tex]\( y \)[/tex] is 3.
- So, the degree of this term = [tex]\( 2 + 3 = 5 \)[/tex].
- For the term [tex]\( xy^2 \)[/tex]:
- The exponent of [tex]\( x \)[/tex] is 1 (assuming it is [tex]\( x^1 \)[/tex]).
- The exponent of [tex]\( y \)[/tex] is 2.
- So, the degree of this term = [tex]\( 1 + 2 = 3 \)[/tex].
- For the constant term [tex]\( 8 \)[/tex]:
- A constant term has a degree of 0.
3. Finding the missing exponent:
- To ensure the expression [tex]\( 5x^2 y^3 + xy^2 + 8 \)[/tex] is a polynomial of degree 5, the highest degree term must have a degree of 5.
- We already see that the term [tex]\( 5x^2 y^3 \)[/tex] has a degree of 5.
- The term [tex]\( xy^2 \)[/tex], as it currently stands, has a degree of 3.
4. Determining necessary conditions:
- We are not altering the first term as it already contributes perfectly to the degree 5 polynomial.
- The second term [tex]\( xy^2 \)[/tex] must not exceed the polynomial degree of 5. Thus, the missing exponent on the [tex]\( x \)[/tex]-term must be chosen carefully to ensure the polynomial does not exceed degree 5.
5. Conclusion:
- The polynomial [tex]\( 5x^2 y^3 + x y^2 + 8 \)[/tex] still satisfies the condition of being a degree 5 polynomial with the highest degree term [tex]\( 5x^2 y^3 \)[/tex].
- A missing exponent should maintain or classify terms under required parameters. Thus, we conclude:
- The missing exponent on the [tex]\( x \)[/tex]-term in [tex]\( xy^2 \)[/tex] must be [tex]\( 1 \)[/tex].
Hence, the missing exponent on the [tex]\( x \)[/tex]-term must be 1 for the expression to be a trinomial with a degree of 5.
