To determine the missing exponent of [tex]\( y \)[/tex] in the polynomial [tex]\( 6xy^2 - 5x^2y^a + 9x^2 \)[/tex] so that it is a trinomial with a degree of 3 after simplification, we need to analyze the terms and their degrees.
1. Let's first break down the polynomial into its individual terms and degrees:
- The first term is [tex]\( 6xy^2 \)[/tex]. The degree of this term is the sum of the exponents of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: [tex]\( 1 \)[/tex] (for [tex]\( x \)[/tex]) + [tex]\( 2 \)[/tex] (for [tex]\( y \)[/tex]) = 3.
- The second term is [tex]\( -5x^2y^a \)[/tex]. The degree of this term is the sum of the exponents of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: [tex]\( 2 \)[/tex] (for [tex]\( x \)[/tex]) + [tex]\( a \)[/tex] (for [tex]\( y \)[/tex]).
- The third term is [tex]\( 9x^2 \)[/tex]. The degree of this term is the exponent of [tex]\( x \)[/tex]: [tex]\( 2 \)[/tex].
2. For the polynomial to be a trinomial with a degree of 3, the highest degree term must have a degree of 3. Currently, the first term [tex]\( 6xy^2 \)[/tex] has a degree of 3, and the third term [tex]\( 9x^2 \)[/tex] has a degree of 2, which is lower than 3.
3. To ensure the polynomial has a degree of 3, we need the second term [tex]\( -5x^2y^a \)[/tex] to also have a degree of 3.
4. Set up the equation for the degree of the second term to equal 3:
[tex]\[
2 + a = 3
\][/tex]
5. Solve for [tex]\( a \)[/tex]:
[tex]\[
a = 3 - 2
\][/tex]
[tex]\[
a = 1
\][/tex]
Therefore, the missing exponent of [tex]\( y \)[/tex] in the second term is [tex]\( 1 \)[/tex].
So, the correct answer is [tex]\( 1 \)[/tex].
