To determine the missing exponent of [tex]\( y \)[/tex] in the second term of the polynomial [tex]\( 6xy^2 - 5x^2y^m + 9x^2 \)[/tex] so that it becomes a trinomial with a degree of 3 after it has been fully simplified, let's proceed with a step-by-step analysis.
1. Identifying the degrees of each term:
- The first term is [tex]\( 6xy^2 \)[/tex]:
- Degree of [tex]\( x \)[/tex] is 1.
- Degree of [tex]\( y \)[/tex] is 2.
- Total degree of this term = [tex]\( 1 + 2 = 3 \)[/tex].
- The third term is [tex]\( 9x^2 \)[/tex]:
- Degree of [tex]\( x \)[/tex] is 2.
- Since there is no [tex]\( y \)[/tex], its degree is 0.
- Total degree of this term = [tex]\( 2 + 0 = 2 \)[/tex].
2. Ensuring the polynomial is a trinomial with a degree of 3:
- Since we need the polynomial to have a degree of 3, the second term must also have a total degree of 3 when simplified.
- The second term is [tex]\( -5x^2y^m \)[/tex]:
- Degree of [tex]\( x \)[/tex] is already 2.
- We need the combined degree to be 3, so we set the equation:
[tex]\[
2 + m = 3
\][/tex]
3. Solving for the missing exponent [tex]\( m \)[/tex]:
- Isolate [tex]\( m \)[/tex]:
[tex]\[
m = 3 - 2
\][/tex]
[tex]\[
m = 1
\][/tex]
Thus, the missing exponent of [tex]\( y \)[/tex] in the second term [tex]\( -5x^2y^m \)[/tex] must be [tex]\( \boxed{1} \)[/tex].
